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LIBRARY 

OF  THE 


UNIVERSITY  OF  CALIFORI^ 


Received      </Jf~~ 
Accession  No.  6   7  J  £<£ 


1 


i 


ECLECTIC    EDUCATIONAL    SERIES. 


JKAY'S  ARITHMETIC,    SECOND  BOOK. 


INTELLECTUAL 

ARITHMETIC 

BY 

INDUCTION  AND  ANALYSIS. 

BY   JOSEPH    RAY,    M.  D., 

LATB  PROFESSOR  OP   MATHEMATICS    IN   WOODWARD   COLLEGE. 

ONE.  THOUSANDTH  EDITION-IMPROVED, 


CINCINNATI  : 
WILSON,     HINKLE     &     OOo 

PHIL'A:  CLAXTON,  REMSEN  &  HAFFELFINGEB. 
NEW  YORK:  CLARK  &  MAYNARD. 


y_ 

RAY'S  MATHEMATICAL  WORKS. 

TYPE   ENLARGED — NEW   ELECTROTYPE   PLATES. 

Each  BOOK  of  Ray  s  Arithmetical  Course,  also  of  the  Algebraic* 
is  a  complete  work  in  itself,  and,  is  sold  separately. 


FIRST    BOOK. 

PRIMARY  LESSONS  AND  TABLES;  simple  and  progressive 
Mental  Lessons,  for  little  learners. 


SECOND    BOOK. 

INTELLECTUAL    ARITHMETIC,   by   Induction    and    Analysis; 
a  thorough  course  on  Intellectual  Arithmetic. 


THIRD    BOO  K. 

PRACTICAL  ARITHMETIC,  by  Induction  and  Analysis ;  a  sim- 
ple, thorough  work  for  schools  and  private  students. 
~J3Y  TO  RAY'S  ARITHMETIC,  THIRD  BOOK. 
RAY'S  HIGHER  ARITHMETIC.    Principles  of  Arithmetic,  ana- 
lyzed and  applied.     For  advanced  students  and  business  men. 

KEY  TO  RAY'S  HIGHER  ARITHMETIC. 


ELEMENTARY    ALGEBRA, 

RAY'S     ALGEBRA,    FIRST    BOOK,    for    Common    Schools    and 
Academies;  a  simple,  progressive,  elementary  treatise. 


HIGHER    ALGEBRA. 

RAY'S   ALGEBRA,  SECOND   BOOK,  for  Academies,  and   for  Col- 
leges ;  a  progressive,  lucid,  and  comprehensive  work. 
KEY  TO  RAY'S  ALGEBRA,  FIRST  AND  SECOND  BOOKS. 

Entered  according  to  Act  of  Congress,  in  the  year  Eighteen  Hundred  and  Fifty- 
Seven,  by  WINTHROP  B.  SMITH,  in  the  Clerk's  Office,  of  the  District  Court  of  the 
United  States,  for  the  Southern  District  of  Ohio. 

Entered  according  to  Act  of  Congress,  in  the  year  Eighteen  Hundred  and  Sixty, 
by  W.  B  SMITH,  in  the  Clerk's  Office  of  the  District  Court  of  the  United  States, 
for  the  Southern  District  of  Ohio. 


PREFACE. 


Few  works  on  Intellectual  Arithmetic  have  received  more 
nnqualified  approbation,  and  a  more  extensive  patronage,  than 
this,  which,  for  several  years,  has  been  published  under  the 
title — Ray's  Arithmetic,  Part  Second. 

The  numerous  editions  demanded,  have  again  rendered 
necessary  a  renewal  of  the  plates,  which  has  afforded  an 
opportunity  for  REMODELING  the  work. 

Many  important  improvements  have  been  introduced,  with 
a  design  to  impart  completeness,  and  give  a  concise  and  pro- 
gressive course  of  arithmetical  analysis. 

The  volume  in  its  present  form,  embraces, 

1st.  Exercises  on  the  primary  principles  and  their  applica- 
tions, interspersed  with  appropriate  models  of  analysis  and 
frequent  reviews. 

2d.  A  progressive  and  comprehensive  presentation  of  Frac< 
tions,  intended  to  render  the  subject  intelligible  and  attractive 
to  the  pupil. 

3d.  A  General  Review,  designed  to  test  the  pupil's  knowledge 
of  principles,  preparatory  to  the  applications  of  mental  analysis 
which  follow. 

4th.  Percentage,  Gain  and  Loss,  Interest,  and  their  appli- 
cations. 

The  value  of  Intellectual  Arithmetic  is  so  highly  appreciated 
by  instructors,  that  little  need  be  said  in  its  commendation. 

When  'properly  taught,  it  is  one  of  the  most  useful  and 
interesting  studies  in  which  pupils  can  engage,  and  should  be 
omitted  by  no  one. 

By  its  study,  learners  are  taught  to  reason,  to  analyze,  ie 
think  for  themselves;  while  it  imparts  co'nfidence  in  their  ows 
reasoning  powers,  and  strengthens  the  mental  faculties. 


HINTS   TO    TEACHERS. 


LET  the  pupils  be  classified  with  reference  to  their  attainments 
and  abilities.  The  recitation  should  be  short  and  spirited,  every 
pupil  being  required  to  give  undivided  attention  to  the  question 
before  the  class. 

Generally,  while  reciting,  the  pupils  should  be  permitted  to  have 
their  books  open  before  them — the  test  of  having  properly  studied 
the  lesson,  being  the  readiness  and  accuracy  with  which  the  several 
questions  are  analyzed  and  answered. 

The  explanations  and  operations  termed  ANALYSIS,  are  intended 
as  Model  Solutions,  pointing  out  to  the  learner  the  manner  in  which 
the  questions  in  the  lessons  are  to  be  solved  and  explained. 

The  pupil  should  be  required  to  furnish  a  similar  explanation 
to  each  of  the  jucceeding  questions,  and  to  give,  not  only  a  cor- 
rect answer,  but  also,  the  reason  for  the  method  by  which  he 
obtained  it. 

A  method  of  solving  questions  in  Mental  Arithmetic,  now  mucb 
used,  is  the  following,  called  the  "Four  Step  Method." 

ILLUSTRATIONS. —  First  step,  James  gave  7  cents  for  apples  and 
8  cents  for  peaches;  how  many  cents  did  he  spend?  Second 
step,  as  many  as  the  sum  of  7  cents  and  8  cents.  Third  step,  1 
cents  and  8  cent1?  are  15  cents.  Fourth  step,  hence,  if  James  gave 
7  cents  for  apples,  and  8  cents  for  peaches,  he  spent  15  cents. 

Again:  First  step,  4-fifths  of  25  are  how  many  times  6?  Second 
ttep,  as  many  times  6  as  6  is  contained  times  in  4-fifths  of  25. 
Third  step,  1-fifth  of  25  is  5,  4-fifths  are  4  times  5,  which  are  20; 
6  in  20  is  contained  3  and  2-sixths  times.  Fourth  step,  therefore, 
4-fifths  of  25  are  3  and  2-sixths  times  6? 

As  a  means  of  keeping  the  attention  of  the  class  directed  to  each 
question,  it.  will  be  proper  for  the  instructor  occasionally  to  read 
an  example  aloud,  and,  having  allowed  sufficient  time  for  the 
answer,  to  call  upon  some  one  for  the  result,  and  then  for  the 
analysis.  By  this  means,  each  one  is  obliged  to  solve  the  example^, 
not  knowing  but  that  he  may  be  required  to  answer  it. 


INTELLECTUAL 


ARITHMETIC. 


SECTION   I.— NUMERATION. 

Pupils  who  have  studied  Ray's  Arithmetic,  First  Book,  maj 
commence  with  Lesson  IV,  on  page  14. 

NUMERATION  is  naming  Numbers. 
Learn  the  name  and  form  of  these  figures: 

1      2      3      4     5     G     7     .8      9      10. 

One,      two,     three,    four,    five,     six,    seven,   eight,   nine,      ten. 


An  Illustration  of  the  Increase  of  Numbers. 


l 

2 
3 
4 
5 

6 

7 

8 

9 

10 


10  KAY'S  INTELLECTUAL  ARITHMETIC. 

SECTION  II.— ADDITION. 

LESSON  I. 

1.  James  had  one   apple,  and   his   brother  gave  him 
one  more  :  how  many  had  he  ?     Why  ? 

Ans.  Because  1  and  1  are  2. 

2.  Henry  had  two  cents,  and  his  sister  gave  him  one 
more  :  how  many  had  he  ?     Why  ? 

Ans.  Because  2  and  1  are  3. 

3.  A  hoy  had  one  marble,    and  found   three   more : 
how  many  did  he  then  have  ?     Why  ? 

Ans.  Because  1  and  3  are  4. 

4.  Thomas  had   4   cents,    and   his   mother  gave   him 
one  more  :  how  many  cents  had  Thomas  ?     Why  ? 

5.  Samuel  had  two  cakes,  and  his  father  gave  him  two 
more  :  how  many  did  he  then  have  ?     Why  ? 

6.  Three   oranges    and    two    oranges   are    how   many 
oranges  ?     Why  ? 

7.  James  had  three  apples,  and  his  brother  gave  him 
three  more  :  how  many  apples  had  James  then  ?     Why  ? 

8.  John  had  four  plums,  and  his  sister  gave  him  two 
more  :  how  many  did  he  then  have  ?     Why  ? 

9.  Daniel  had  three  cents  ;  his  brother  gave  him  two, 
and  his  sister  one  :  how  many  did  he  then  have?     Why? 

10.  Mary  had  four  pears,   and    her   brother  gave  her 
three  more :  how  many  did  she  then  have  ?     Why  ? 

11.  George   recites    daily    four   lessons    perfectly,    one 
imperfectly,  and  is  absent  from  one  recitation  :  how  many 
daily  lessons  has  he  ?     Why  ? 

12.  How  many  fingers  have  you  on  one  hand  ?     How 
many  on  both?     Why? 

13.  Ida  had  four  cents ;  her  mother  gave  her  three  more 
at  one  time,  and  one  at  another :  how  many  did  she  then 
have?     Why? 


ADDITION.  1 1 

14.  Three  cakes,  and  one  cake,  and  two  cakes,  are  how 
many,  cakes  ?     Why  ? 

15.  Four  cents,  and  three  cents,  and  two  cents,  are  how 
many  cents  ?     Why  ? 

16.  Five    oranges,    and    one    orange,    are    how    many 
oranges  ?     Why  ? 

17.  Henry  had  five  cents,  and  his  mother  gave  him  two 
more:  how  many  did  he  then  have?     Why? 

18.  Five  boys,  and  one  boy,  and  two  boys,  are  hoi 
many  boys  ?     Why  ? 

19.  Oliver  has  five  dollars,    Henry  three,  and  Samuel 
one  :  how  many  dollars  have  all  together  ?     Why  ? 

20.  Three   peaches,  and   six  peaches,    are   how  many 
peaches  ?     Why  ? 

21.  A  lady  paid  one  dollar  for  gloves,  three  dollars  fo>* 
a  shawl,  and  five  dollars  for  a  dress :  how  much  did  si 
spend?     Why? 

22.  Four  cents,  and  three   cents,   and   two  cents,  and 
one  cent,  are  how  many  cents  ?     Why  ? 

23.  If   a  man  buy  six  pounds   of  sugar  at  one  time, 
and  four  at  another,  how  much  does  he  buy  ?     Why  ? 

24.  Seven  oranges,  and  one  orange,  and  two  oranges, 
are  how  many  ?     Why  ? 

25.  George  has  two  cents,  his  sister  three   cents,  and 
his  brother  five  cents :  if  they  were  all  given  to  George 
how  many  would  he  have  ?     Why  ? 

26.  How  many  are  4  and  3  and  3? 

27.  James  has  4  cents,  and  Joseph  2  more  than  James: 
how  many  has  Joseph  ?     How  many  have  both  ?     Why  ? 

LESSON  II. 

1.  One  and  1  are  how  many?  1  and  2?  3  and  1? 
4  and  1?  1  and  3?  1  and  5?  1  and  6?  6  and  1? 
1  and  7?  8  and  1?  9  and  1  ?  1  and  8? 


12  RAY'S   INTELLECTUAL  ARITHMETIC. 

2.  Two  and  2  are  how  many?     2  and  3?     2  and  2 
and  1?     1  and  2  and  3?     4  and  1  and  2? 

3.  Four  and  2  are  how  many  ?     4  and  3  ?      2  and  4 
andl?     4  and  land  2?     3  and  4?     5  and  4? 

4.  Six  and  2  are  how  many?      6  and  1  and  2?      6 
and  3?      6  and  4?     6  and  1  and   3?     1  and  2  and  6? 

1  and  3  and  6? 

5.  Eight  and  2  are  how  many?      8  and  3?      8  and 

2  and  1?      2  and  8  and   1  ?      1  and  4  and  3?      8  and 
4?      7  and  4?      6  and  7?     7  and  1  and  3*      1  and  6 
and  2  ?     1  and  8  and  2  and  2  ? 

6.  Nine  and  2  are  how  many?      9  and  3?      9  and 

1  and  2?      8  and  1  and  3  ?      5  and  6  ?      1  and  4  and 
5?      2  and  4  and  6?      3  and  4  and  5?      4  and  6  and 

2  ?     5  and  4  and  2  and  1  ? 

7.  How  many  are  1  and  9?      7  and  3?      4  and  6? 
9  and  1?     5  and  5?     6  and  4?     2  and  8  ?     3  and  7? 
0  and  10?     8  and  2?     10  and  0?     What  two  numbers 
added  together  make  10? 

8.  Which  is  the  greater,  7  and  2,  or  6  and  4?     Why? 
one  and  3  and  6,  or  8  and  0  and  2  ?     Why  ? 

9.  Begin  at  4,  and  add  2  each  time,  up  to  10. 

10.  Begin  at  1,  and  add  3  each  time,  up  to  13. 

11.  What  two  numbers  added  together  will  make  12? 
What  three  numbers? 

12.  Seven  and  5  and  2  are  how  many? 

13.  One  and  7  and  3  and  4  are  how  many? 

14.  If  3  be  added  to  3,  and  that  sum  to  4,  what  will  be 
the  result? 

15.  Tf  you  add  3,  to  3  and  1  more,  and  then  add  7, 
what  will  be  the  amount? 

16.  I  have  in  one  basket  8  dozen  eggs,  in  another  4 
dozen,  in  another  3  dozen:  how  many  in  all? 


ADDITION.  13 

17.  A  lady  bought  2  yards  of  tape  for  3  cents,  some 
pins  for  1  cent,  and  received  2  cents  change  :  how  many 
cents  had  she  at  first? 

18.  Two  and  1  more,  and  3  and  4  more,  are  together 
how  many  ? 

10.  One  and  3  and  4  and  5  are  how  many?      5  and 
one  and  3  and  4  ? 

20.  Two  and  1   and  3,  taken   from  a  certain  number, 
leave  2  :  what  is  that  number  ? 

21.  A  boy  bought  3   cents   worth  of  marbles,  and  2 
cents  worth  of  candy,  and  received  5  cents  change :  how 
muck  money  had  he  ? 

22.  I  bought  3  hams  for  8  dollars,  and  10  bushels  of 
apples  for  3  dollars :  how  much  did  I  spend  ? 

23.  If  6  yards  of  cloth  will  make  2  coats,  and  4  yards 
will  make  2  pairs  of  pants,  how  many  yards  must  I  buy 
for  2  coats  and  2  pairs  of  pants? 

24.  Oliver  has   4  cents  in  one  hand,  3  in  the   other, 
and  4  in  his  pocket :  how  many  cents  has  he  ? 


LESSON  III— ADDITION  TABLE. 


2    and       0    are       2 

O    and        U    are       O 

4    and        0    are       4 

2    and       1    are       O 

3    and        1    are       4 

4    and       1    are        5 

2    and       2    ,re       4 

3    and        2    are        5 

4    and       2    are       6 

2    and       3    are       5 

3    and        3    are       6 

4    and        3    are        7 

2    and       4    are       6 

8    and       4    are       7 

4    and        4    are        8 

2    and       5    are       7 

8    and        5    are       8 

4    and        5    are       9 

2    and       6    are        8 

3    and       6    are       9 

4    and       6    are    10 

2    and       7"    are       9 

3    and        7    are    10 

4    and        *7    are    H 

2    »nd        8    are    10 

3    and        8    are    11 

4    and        8    are    12 

2    and        9    are    H 

3    and       9    are    12 

4    and        9    are    13 

2    and    10    are    12 

3    and    10    are    13 

4    and    10    are    14 

9    and    11    a,e    13 

3    and    11    are    14 

4    and    11    are    15 

2    and    12    are    14 

3    and    12    are    15 

4   and    12    are    16 

14 


RAY'S   INTELLECTUAL   ARITHMETIC. 


5    and       0    are        5 

6    and       0    are       6 

7    and       0    are       7 

5    and       1    are       6 

6    and       1    are       7 

7    and       1    are       8 

5    and       2    are        7 

6    and       2    are       8 

7    and       2    are       9 

5    and       3    are       8 

b    and       3    are       9 

7    and       3    are    10 

5    and       4    are       9 

6    and       4    are    10 

7    and       4    are    11 

5    and       5    are    10 

6    and       5    are    H 

7    and       5    are    12 

5    and       6    are    H 

6    and       6    are    12 

7    and       6    are    13 

5    and       7    are    12 

6    and       7    are    13 

7    and       7    are    14 

5    and       8    are    13 

6    and       8    are    14 

7    and       8    are    15 

5    and       9    are    14 

6    and       9    are    15 

7    and       9    are    16 

5    and    10    are    15 

6    and    10    are    16 

7    and    10    are    17 

5    and    11    are    16 

6    and    11    are    17 

7    and    11    are    18 

5    and    12    are    17 

6    and    12    are    18 

7    and    12    are    19 

8    and       0    are        O 

9    and       0    are       9 

10    and       0    are    10 

8    and       1    are       9 

9    and       1    are    10 

10    and       1    are    11 

8    and       2    are    10 

9    and       2    are    11 

10    and       2    are    12 

8    and        3    are    11 

9    and       3    are    12 

.10    and       3    are    13 

8    and        4    are    12 

9    and       4    are    13 

10    and       4    are    14 

8    and        5    are    13 

9    and       5    are    14 

10    and       5    are    15 

8    and       6    are    14 

9    and       6    are    15 

10    and       6    are    16 

8    and       7    are    15 

9    and       7    are    16 

10    and       7    are    17 

8    and       8    are    16 

9    and        8    are    17 

10    and       8    are    18 

8    and        9    are    17 

9    and       9    are    18 

10    and       9    are    19 

8    and    10    are    18 

9    and    10    are    19 

10    and    10    are    20 

8    and    11    are    19 

9    and    11    are    20 

10    and    11    are    21 

8   and    12    are    20 

9    and    12    are    21 

10    and    12    are    22 

LESSON  IV. 

NOTE. — In  the  following  exercises  let  the  numbers  be  added 
both  ways:  thus,  2  and  8  are  5,  3  and  2  are  5;  4  and  3  are 
7,  3  and  4  are  7 ;  etc.  In  adding  such  numbers  as  16  and  12 ; 
say  16  and  10  are  26,  and  2  are  28. 

1.  Three  and  8  are  how  many?     3  and  10? 

2.  Four   and  7  are   how  many?     4   and    9?     4   and 
eleven?     4  and  10?     4  and  12? 


ADDITION.  15 

3.  Five  and  7  are  how  many?     5  and  6?     5  and  9? 
five  and  12?     5  and  10?     5  and  8  ?     5  and  11? 

4.  Seven  and  6  are  how  many?     7  and  10?     7  and 
eight?     7  and  12?     7  and  9  ?     7  and  11? 

5.  Nine  and  11  are  how  many  ?     9  and  9  ?     9  and  12  ? 
nine  and  7  ?     9  and  10  ?     9  and  8  ?     9  and  11  ? 

6.  Ten  and  6  are  how  many?     10  and  8?     10   and 
ten?     10  and  12?     10  and  11?     10  and  5?     10  and  7? 
ten  and  9  ? 

7.  Eleven  and  2  are  how  many  ?     11  and  4?     11  and 
six?     11  and  5?     11  and  7?     11  and  9?     11  and  11? 
eleven  and  10? 

8.  Twelve  and  5  are  how  many?      12  and  4?      12 
and  6?      12    and   8?      12   and    10?      12  and  7?      12 
and  9?      12  and  11?      12   and    12? 

9.  Thirteen    and    4    are    how    many  ?      13   and    6  ? 
13    and    5?      13    and    7  ?      13    and    9?      13   and  10  ? 

13  and    8?      13  and  11?      13  and  12? 

10.  Fourteen  and  four  are  how  many?      14   and    6? 

14  and    8?      14    and    5  ?      14    and    7?      14   and  10? 
14    and    9?      14    and  11  ?      14    and  12  ? 

11.  Fifteen  and  5  are  how  many?      15  and  7?      15 
and  9?     15  and  4?     15  and  8?     15  and   10?     15  and 
12?     15  and  11? 

12.  Sixteen  "and  4  are  how  many?      16  and  6?      16 
*nd  8?     16  and  5?     16  and  7?     16  and  9?     16  and  11? 
16  and  10  ?     16  and  12  ? 

13.  Seventeen  and  6  are  how  many?     17  and  4?     17 
a^d  7  ?      17  and  5  ?     17  and  9  ?      17  and  8  ?     17  and 
10  ?     17  and  12  ?     17  and  11  ? 

14.  Eighteen  and  10  are  how  many?     18  and  4?     18 
and  7?     18  and  5?     18  and  8?     18  and  6?     18  and  9? 

18  and  11  ?     18  and  12  ? 

15.  Nineteen  and  5  are  how  many?     19  and  3?     19 
and  2?     19  and  7?     19  and  9?     19  and  8?     19  and  10? 

19  and  6?     19  and  12  ?     19  and  11? 


16  RAY'S    INTELLECTUAL  ARITHMETIC. 

16.  How  many  are  9  and  2  ?     19  and  2?     29  and  21 
49  and  2  ?     6(J  and  2  ?     39  and  2  ?     59  and  2  ?     79  and 

2  ?     99  and  2  ? 

17.  How  many  are  9  and  3?     3  and  19?     29  and  3? 

3  and  4(J  ?     5U  and  3?     3  and  39?     69  and  3?     3  and 
79?     3  and  89?     99  and  3? 

18.  How  many  are  9  and  7  ?     29  and  7  ?     7  and  49? 
39  and  7  ?     7  and  59  ?     79  and  7  ?     7  and  69  ?     89  and 
7?     7  and  99? 

19.  How  many  are  9  and  8?     29  and  8?     49  and  8? 
39  and  8?     8  and  69?  .  59  and  8?     79  and  8? 

20.  How  many  are  9  and  9?     19  and  9?     9  and  29? 
49  and  9?     69  and  9?     59  and  9?     79  and  9?     89  and 
9?     9  and  99? 

21.  How  mar-  are  8  and  3  ?     28  and  3?     48  and  3? 
68  and  3?     88  and  3?     98  and  3? 

22.  How  many  are  8  and  7  ?     28  and  7  ?     7  and  38  ? 
48  and  7?    68  and  7  ?     58  and  7?    88  and  7? 

23.  How  many  are  7  and  7  ?     17  and  7?     27  and  7? 
47  and  7  ?     57  and  7  ?     37  and  7  ?    67  and  7  ?     87  and 
7  ?     77  and  7  ?     97  and  7  ? 

24.  How  many  are  7  and  10?     17  and  10?     27  and 
10  ?     47  and  10  ?     37  and  10  ?     57  and  10  ? 

25.  How  many  are  6  and  5  ?     16  and  5?     15  and  6? 
26  and  5?     25  and  6?     24  and  6?     26  and  4?     36  and 
6?     48    and    6?     45    and   6?     57    and   6?     59  and   6? 
66  and  6?     75  and  6?     86  and  6? 

26.  How  many  are  13  and  8?     17  and  3  ?     23  and  8? 
24  and  8?     33  and  8  ?     3  and  37?     8  and  43?     47  and 
3?     7  and  53?     58  and  3  ?     67  and  3?     3  and  87  ?     97 
and  3?     88  and  3?     3  and  98  ? 

27.  How  many  are  14  and  9?     9  and  24?     25  and  9? 
9  and  34  ?     36  and  9  ?     9  and  44  ?     9  and  47  ?     54  and 
9?     9  and  56?     9  and  64?     74  and  9?     9  and  72?     8* 
and  9  ?     86  and  9  ?     94  and  9  ? 


ADDITION.  17 

28.  How   many  are    10  and  6?     6  and  21  ?     10  and 
26?     46  and  10?     10  and  35?     10    and   55?     56    and 
10?     10  and  66?     10  and  69?     76   and    10?     10    and 
86?     96  and  10? 

29.  How  many  are  11  and  6?     11  and  16?     11  and 
27?      25  and  11?      11  and  23?      31  and  11?     11  and 
35?      37  and  11?      11  and  59?      46  and  11?     11  and 
48?      52  and  11?      11  and  63? 

30.  Fifty-six  and  1  are   how  many?     56  and  3?     56 
and  5?     2  and  56?     4  and  56?     7  and  56  ?     56  and  6? 
8  and  56  ?     56  and  9  ?     10  and  56  ? 

31.  Ninety-eight  and  1  ?     3  and  98?     5  and  98?     98 
and  2  ?     4  and  98  ?     98  and  6  ?     7  and  98  ?     98  and  8  ? 
10  and  98?     98  and  9? 

32.  How  many  are  1  and  2  and  4  and  5  ?     5  and  1  and 
four  and  2  ?     2  and  1  and  5  and  4  ? 

33.  How  many  are  12  and  8  ?     9  and  2  and  9?     4  and 
2  and  9  and  5  ?     4  and  2  and  5  and  5  and  4  ?     8  and  4 
and  6  and  2  ?     8  and  9  and  6  ? 

34.  If  you  add  3  to  3,  and  4  to  4,  and  5  to  5,  and  add 
those  sums  together,  what  number  will  you  have  ? 

35.  If  to  10  you  add  6,  7,  and  9,  how  much  will  you 
then  have  ? 

LESSON   V. 

NOTE. — The  numbers  in  the  following  examples  should  be 
added  aloud;  and  if  preferred,  the  "Four  step  method"  can  be 
applied  to  all  the  questions. 

Take  for  example  the  second  question ;  Four  and  5  and  7  are 
how  many?  As  many  as  the  sum  of  4  and  5  and  7;  4  and  5 
are  9,  9  and  7  are  1C ;  therefore,  4  and  5  and  7  are  16. 

1.  Three  and  6  and  4  are  how  many  ? 

2.  Four  and  5  and  7  are  how  many  ? 

3.  Five  and  6  and  2  are  how  many  ? 

4.  Six  and  4  and  5  are  how  many? 


18  RAY'S   INTELLECTUAL   ARITHMETIC. 

5.  Seven  and  3  and  5  and  2  are  how  many  ? 

6.  Eight  and  2  and  3  and  4  are  how  many  ? 

7.  Nine    and  2  and  4  and  3  are  how  many  ? 

8.  Two     and  9  and  5  and  4  are  how  many  ? 

9.  Three  and  9  and  5  and  4  are  how  many  ? 

10.  Four  and  8  and  3  and  5  and  2  and  6  and  3  and 
one  are  how  many  ? 

11.  Five  and  7  and  2  and  3  and  4  and  6  and  5  and  2  are 
how  many  ? 

12.  Two  and  4  and  3  and  5  and  6  and  2  and  7  and  4  are 
how  many  ? 

13.  Three  and  2  and  4  and  5  and  4  and  6  and  3  and 
seven  and  5  are  how  many  ? 

14.  Four  and  3  and  5  and  7  and  6  and  8  and  2  and  4  are 
how  many  ? 

15.  Four  and  9  and  3  and  5  and  6  and  7  and  8  and  9  are 
how  many  ? 

16.  Five  and  8  and  5  and  8  and  5  and  8  and  5  and  8  are 
how  many  ? 

17.  Six  and  8  and  7  and  3  and  5  and  4  and  7  and  1  and 
nine  are  how  many  ? 

18.  Seven  and  9  and  5  and  4  and  6  and  3  and  8  and 
five  and  9  are  how  many  ? 

19.  Eight  and  7  and  6  and  5  and  4  and  9  and  3  and 
seven  and  8  are  how  many  ? 

20.  Nine  and  6  and  7  and  4  and  5  and  3  and  8  and 
two  and  9  are  how  many  ? 

21.  Seven  and  6  and  5  and  8  and  7  and  9  and  8  and 
four  and  9  and  8  are  how  many  ? 

22.  Nine   and  8  and  7  and  5  and  8  and  9  and  5  and 
four  and  7  and  3  and  9  and  8  are  how  many  ? 

23.  Twelve  and  11  and  7  and  4  and  9  are  how  many  ? 

24.  Thirteen  and  10  and  8  and  6  and  4  and  10  are  how 
many? 


ADDITION.  19 

25.  Fourteen  and  16  and  7  and  5  and  9  and  8  and  9  and 
Bix  and  4  are  how  many  ? 

26.  James   gave   7   cents  for  apples,   and  8   cents  for 
peaches :  how  many  cents  did  he  spend  ? 

27.  Seven  dollars,  and  5  dollars,  and  3  dollars,  are  how 
many  dollars  ? 

28.  David  had  11  books ;  he  bought  7  more,  and  his 
brother  gave  him  5 ;  how  many  had  he  then  ? 

29.  A  man  gave  13  dollars  for  a  cart,  6  for  a  plow,  and 

5  for  a  harrow :  how  many  dollars  did  he  spend  ? 

30.  James  has  8  marbles  in  one  pocket,  5  in  another^ 

6  in  another,  and  7  in  another  :  how  many  in  all  ? 

31.  If  a  pound  of  butter  cost  18  cents,  of  lard  7  cents, 
and  of  cheese  9  cents,  how  much  will  all  cost? 

32.  A  man   owes  to  one  person  8  dollars,  to  another 
5  dollars,  to  another  3  dollars,  and  to  another  7  dollars: 
how  much  does  he  owe  ? 

33.  A  boy  gave  19  cents  for  a  spelling-book,  8  cents 
for  a  slate,  and  6  cents  for  pencils :  how  many  cents  did 
he  spend? 

34.  A  drover  bought  hogs  as  follows :  of  one  man  17, 
of  another  9,  of  another  7,  of  another  8 :  how  many  did 
he  buy  ? 

35.  A  little  girl  gave  10  cents  for  thread,  7  cents  for 
pins,   6  cents  for  needles,   and   9   cents   for  tape  :    how 
many  cents  did  she  spend  ? 

36.  William   has  7  cents,  Thomas  10  cents,   David  9 
cents,    and  Moses  8  cents:   if   the  other  boys  give  their 
cents  to  Moses,  how  many  will  he  have? 

37.  The   age  of  Thomas  is  8  years ;  Frank,  5  years ; 
and  William  is  as  old  as  both  together :  what  "is  the  sum 
of  all  their  ages? 

38.  Joseph  has  4  marbles,  William  has  2,  and  David 
has  twice  as  many  as   Joseph  :    how  many  do  they  all 
have? 


20  RAY'S   INTELLECTUAL   ARITHMETIC. 

39.  Begin  with  2,   and  count  one  hundred  by  adding 
2  successively.     Thus,  2  and  2  are  4,  and  2  are  6,  and  2 
are  8,  and  2  are  10,  and  so  on. 

40.  Begin  with  3,  and  count  ninety-nine  by  adding  3 
successively.     Thus,  3  and   3  are  6,  and  3  are  9,  and  3 
are  12,  and  so  on. 

41.  Begin  with  4,  and  count  one   hundred  by   adding 
four  successively. 

42.  Begin  with  5,  and  count  one  hundred  by  adding 
five  successively. 

43.  Begin  with  6,  and  count  one  hundred  and  two  by 
adding  6  successively. 

44.  Begin   wit1    7,  and   count  ninety-eight  by  adding 
seven  successive^ . 

45.  Begin  with  8,  and  count  one  hundred  and  four  by 
adding  4  successively. 

46.  Begin   with  9,   and   count   ninety-nine    by  adding 
nine  successively. 

47.  Begin  with  1,  and  count  one  hundred  by  adding 
three  successively. 

48.  Begin  with  3,   and  count   one   hundred  and  three 
by  adding  4  successively. 

49.  Begin  with  2,  and  count  one  hundred  and  two  by 
adding  5  successively. 

50.  Begin  with  5,  and  count  one   hundred    and  seven 
by  adding  6  successively. 

51.  Begin  with  6,  and  count  one  hundred  and  four  by 
adding  7  successively. 

52.  Begin  with  7,  and  count  one  hundred  and  three  by 
adding  8  successively. 

53.  Begin  with  8,  and  count  one  hundred  and  seven  by 
adding  9  successively. 

54.  If  an  apple  cost  3  cents,  and  an  orange  cost  2  cents 
more    than   an  apple,  what  will  be  the  cost  of  2  apples 
and  one  orange? 


ADDITION.  21 

55.  James  has  5  marbles ;  Henry,  2  more  than  James ; 
and   Samuel,  as  many  as  both  James  and   Henry  :    how 
many  have  all  ? 

56.  Mary  had  a  certain  number  of  peaches ;  she  gave 
5  to   her   sister,  7  to   her  brother,  and  then  had  10  re- 
maining :  how  many  had  she  ? 

57.  A  boy  bought  a  sled  for  20  cents,  and  paid  5  cents 
for  repairing  it:  what  was  it  then  worth? 

53.  If   a  hat   cost  4  dollars,   and  a  coat  as  much  as 
three  hats,  what  do  they  both  cost? 

59.  How  many  times  must  10  be  added  to  fifteen,  to 
make  fifty-five? 

60.  A  boy's  father  gave  him  5  cents ;  his  mother,  one 
cent  more   than  his   father ;  and   his   brother,   two  cents 
more  than  his  father:  how  many  did  he  then  have  ? 

61.  What  is  the  exercise  of  putting  numbers  together 
called?     Ans.  Addition. 

62.  What  is  simple  Addition? 

Ans.   Simple  Addition   is  finding  the    SUM  of   two  or 
more  numbers  of  the  same  denomination. 

63.  What  is  meant  by  same  denomination  f 

Ans.  Same  denomination  means  all  of  the  same  name; 
that  iSj  all  pounds,  or  all  dollars,  &c. 


SECTION  III.— SUBTRACTION. 

LESSON  I. 

1.  James  had  2  apples,  and  gave  1  to  his  brother: 
how  many  had  he  left?     Ans.  1.     Why?     Because  1  and 
I  are  2. 

2.  Then  1  from  2  leaves  how  many  ? 

3.  Jeseph  had  3  apples  and  lost  1  :  how  many  had  lia 
Jeft  ?     Ans.  2.     Why  ?     Because  1  and  2  are  3. 

2d  Bk.  2 


22  KAY'S   INTELLECTUAL   ARITHMETIC, 

4.  Then  1  from  3  leaves  how  many? 

5.  Thomas  had  4  cents,  and  gave  1   to  Frank:  hoi* 
many  had  he  left?     Why? 

6.  Then  1  from  4  leaves  how  many  ? 

7.  One  from  5  leaves  how  many?     From  6?    7?    8? 
9?*10? 

8.  John  had  4  cents  and  gave  his  sister  2 :  how  many 
had  he  left?     Why? 

9.  Then  2  from  4  leaves  how  many  ? 

10.  James  had  5  apples,  and  gave  his  brother  2 :  how 
many  had  he  left  ?     Why  ? 

11.  Then  2  from  5  leaves  how  many? 

12.  Two  from  6  leaves  how  many?     From  7?    8?    9? 
10?     11? 

13.  Thomas  had  5  cents  and  lost  3 :  how  many  had  he 
left?     Ans.2.     Why? 

14.  Then  3  from  5  leaves  how  many  ? 

15.  Three  from  6  leaves  how  many?     From  7?     8? 
9?     10?     11?     12? 

16.  Joseph  had  9  marbles-  and  lost  4:  how  many  had 
he  left  ?     Why  ? 

17.  Then  4  from  9  leaves  how  many  ? 

18.  Four  from  10  leaves  how  many?     From  11  ?     12? 
13?     14?     15? 

19.  William  had  10  apples  and  gave  Joseph  5:     how 
many  had  he  left  ?     Why  ? 

20.  Then  5  from  10  leaves  how  many  ? 

21.  Five  from  11  leaves  how  many  ?     From  12?     13? 
14?     15?     16? 

22.  James  had  11  marbles  and  lost  6 :  how  many  had 
he  left?     Why? 

23.  Then  6  from  11  leaves  how  many  ? 


SUBTRACTION.  23 

24.  Six  from  12  leaves  how  many?     From  13?     14? 
15?     16?     17?     18? 

25.  William  had  12  cents  and  lost  7 :  how  many  had 
he  left?     Why? 

26.  Then  7  from  12  leaves  how  many? 

27.  Seven  from  13  leaves  how  many?     From  14?    15? 
16?     17?     18?     19? 

28.  James  had  13  apples  and  gave  his  sister  8 :    how 
many  had  he  left?     Why? 

29.  Then  8  from  13  leaves  how  many? 

30.  Eight  from  14  leaves  how  many  ?     From  15  ?     16? 
17?     18?     19?     20? 

31.  Thomas  had  13  apples  and  gave  his  sister  9:  how 
many  had  he  left  ?     Why? 

32.  Then  9  from  13  leaves  how  many? 

33.  Nine  from  14  leaves  how  many?     From  15?     16? 
i7?     18?     19?    20? 

34.  Henry  had  14  cents  and  lost  5 :  how  many  had  he 
remaining  ? 

35.  Mary  is  15  years  old,  and  Anna  is  8  :  how  much 
older  is  Mary  than  Anna  ? 

36.  Sold  a  load  of  corn  for  17  dollars;  received  for  it 
a  barrel  of  flour  worth  6  dollars,  and  the  rest  in  money : 
how  much  money  did  I  receive  ? 

37.  A  boy  had  18  marbles  and  lost  10  :  how  many  had 
he  then  ? 

38.  Nineteen  is  11  more  than  what  number  ? 

39.  Went  shopping  with  20  dollars ;  spent  10  dollars  in 
one  store,  and  5  in  another :  how  much  had  I  left  ? 


In  the  present  edition  of  this  volume,  a  very  few  examples 
have  been  modified,  the  better  to  adapt  them  to  a  more  perfect 
gradation.  Such  are  indicated  by  the  character,  0.  The  omission 
of  these  examples  in  classes  having  books  of  the  present  and  for* 
mer  edition,  will  obviate  all  confusion. 


24 


RAY'S   INTELLECTUAL    ARITHMETIC. 


LESSON   II.—  SUBTRACTION   TABLE. 


1  from 

1 

leaves 

0 

2  from      2 

eaves 

0 

3 

from 

3 

leaves 

0 

1  from 

2 

leaves 

1 

2  from      3 

eaves 

1 

3 

from 

4 

leavea 

1 

1   from 

3 

leaves 

2 

2  from      4 

eaves 

2 

3 

from 

5 

leaves 

2 

1  from 

4 

leaves 

3 

2  from      5 

eaves 

3 

3 

from 

6 

leaves 

3 

1  from 

5 

leaves 

4 

2  from      6 

eaves 

4 

3 

from 

7 

leaves 

4 

1  from 

6 

leaves 

5 

2  from      7 

eaves 

5 

3 

from 

8 

leaves 

5 

1  from 

7 

leaves 

6 

2  from       g 

eaves 

6 

3 

from 

9 

leaves 

6 

1   from 

8 

leaves 

7 

2  from       9 

eaves 

7 

3 

from 

10 

leaves 

7 

1  from 

9 

leaves 

8 

2  from    10 

leaves 

8 

3 

from 

11 

leaves 

8 

1  from 

10 

leaves 

9 

2  from    11 

leaves 

9 

3 

from 

12 

leaves 

9 

1  from 

11 

leaves 

10 

2  from    12 

leaves 

10 

3 

from 

13 

leaves 

10 

1  from 

12 

leaves 

11 

2  from    13 

leaves 

11 

3 

from 

14 

leaves 

11 

4  from 

4 

leaves 

0 

5   from      5 

leaves 

0 

6 

from 

6 

leaves 

0 

4  from 

5 

leaves 

1 

5   from      6 

leaves 

1 

6 

from 

7 

leaves 

1 

4  from 

6 

leaves 

2 

5   from      7 

leaves 

2 

6 

from 

8 

leaves 

2 

4  from 

7 

leaves 

3 

5  from      8 

leaves 

3 

6 

from 

9 

leaves 

3 

4  from 

8 

leaves 

4 

5  from      9 

leaves 

4 

6 

from 

10 

leaves 

4 

4  from 

9 

leaves 

5 

5  from  10 

leaves 

5 

6 

from 

11 

leaves 

5 

4  from 

10 

leaves 

6 

5  from  11 

leaves 

6 

G 

from 

12 

leaves 

6 

4  from 

11 

leaves 

7 

5   from  12 

leaves 

7 

6 

from 

13 

leaves 

7 

4  from 

12 

leaves 

8 

5  fr01*  13 

leaves 

8 

6 

from 

14 

leaves 

8 

4  from 

13 

leaves 

9 

5  from  14 

leaves 

9 

6 

from 

15 

leaves 

9 

4  from 

14 

leaves 

10 

5  from  15 

leaves 

10 

6 

from 

16 

leaves 

10 

4  from 

15 

leaves 

11 

5   from   16 

leaves 

11 

6 

from 

17 

leaves 

11 

7  from 

7 

leaves 

0 

8-                   0 
irom      O 

leaves 

0 

9 

from 

9 

leaves 

0 

7   from 

8 

leaves 

1 

8  from      9 

leaves 

1  ) 

9 

from 

10 

leaves 

1 

7  from 

9 

leaves 

2 

8  from  10 

leaves 

2 

9 

from 

11 

leaves 

2 

7  from 

10 

leaves 

3 

8  from  11 

leaves 

3 

9 

from 

12 

leaves 

3 

7   from 

11 

leaves 

4 

8  from  12 

leaves 

4 

9 

from 

13 

leaves 

4 

7   from 

12 

leaves 

5 

8  from  13 

leaves 

5 

9 

from 

14 

leaves 

5 

7   from 

13 

leaves 

6 

8  from  14 

leaves 

6 

9 

from 

15 

leaves 

6 

7  from 

14 

leaves 

7 

8  from   15 

leaves 

7 

9 

from 

16 

leaves 

7 

7   from 

15 

leaves 

8 

8  from   16 

leaves 

8 

9 

from 

17 

leaves 

8 

7   from 

16 

leaves 

9 

8  from   17 

leaves 

9 

9 

from 

18 

leaves 

9 

7   from 

17 

leaves 

10 

8  from  18 

leaves 

10 

9 

from 

19 

leaves 

10 

7  from 

18 

leaves 

11 

8  from  19 

leaves 

11 

9 

from 

20 

leaves 

11 

SUBTRACTION.  25 

To  TEACHERS. — Instead  of  requiring  pupils  to  recite  the  Sub- 
traction Table  regularly,  some  of  the  best  instructors  omit  it,  and 
connect  the  exercises  with  addition,  thus: 

1  and  2  are  3  2  and  3  are  5 

2  and  1  are  3  3  and  2  are  5 

1  from  3  leaves  2          2  from  5  leaves  3 

2  from   3  leaves  1          3  from  5  leaves  2 

In  case  the  Table  should  not  be  omitted,  the  above  exercises 
tfill  be  found  very  profitable. 

1.  Two  from  7  leaves  how  many? 


2  from  12? 

3  from     8? 

3  from  10? 

4  from     9  ? 

4  from  12  ? 

4  from  13  ? 

5  from  14? 

5  from  17? 

6  from  11  ? 

6  from  15? 

6  from  18? 

7  from  11  ? 

7  from  15  ? 

7  from  17? 

7  from  19? 

8  from  13? 

8  from  15  ? 

8  from  17  ? 

9  from  13? 

9  from  15? 

9  from  17? 

2.  What  number  must  be  added  to  8  to  make  10  ? 
To  12  to  make  15?  To     9  to  make  15? 

To  17  to  make  20?  To  16  to  make  20? 

To  14  to  make  19?  To  19  to  make  21? 

To  13  to  make  16?  To  12  to  make  20? 

To  11  to  make  20  ?  To  13  to  make  20  ? 

LESSON    III. 

1.  A  boy  gave    9   cents   for  a   spelling-book,   worth 
only  7  cenis  :  how  much  did  he  pay  for  it  more  than  it 
was  worth?     Why? 

2.  A  inan  having  16  dollars  lost  12 :  how  many  had 
he  left?     Why? 

3.  Bought  a  book  for  12  cents,  and  a  top  for  7  cents: 
how  much  did  the  book  cost  more  than  the  top  ? 


26  RAY'S   INTELLECTUAL  AKITHMETIC. 

4.  Thomas  had  18  cents  given  him  by  two  boys ;  one 
gave  9  :  how  many  did  the  other  give  ? 

5.  Bought  a  book  for  14  cents,  and  gave  the  shop- 
keeper 20  cents :  how  much  change  did  he  return  me  ? 

6.  William  has  19  apples ;  in  one  pocket  he  has  15; 
how  many  are  in  the  other  ? 

7.  A  man  has  25  miles  to  travel :  when  he  has  gone 
nineteen,  how  many  will  he  still  have  to  travel  ? 

8.  A  boy  gave  24  cents  for  a  book,  and  sold  it  for 
sixteen  cents :  how  much  did  he  lose  ? 

9.  James  had  24  marbles;  he  gave  19  to  his  brother: 
how  many  had  he  left  ? 

10.  A  man  bought  a   horse   for  19  dollars,  and  sold 
him  for  27  dollars  :  how  much  did  he  gain  ? 

11.  A  man  owing  26  dollars,  paid  18 :  how  many  did 
he  then  owe  ? 

12.  Frank   had   26  cents  given  him   by  William    and 
Thomas.     William  gave  him  17 :  how  many  did  Thomas 
give  ?     How  many  more  did  William  give  than  Thomas  ? 

13.  If  you  had  10  apples,  and  should  give  2  to  John, 
and  6  to  your  sister,  how  many  would  you  have  left? 

14.  Abel    had    36    cents,    and   his    mother   gave   him 
pnough  to  make  40  :  how  many  did  she  give  him  ? 

15.  George   had   40   marbles;   he  lost  20,  and  found 
five :  how  many  did  he  then  have  ? 

16.  A  man  had  100  barrels  of  flour;  he  sold  50,  and 
afterward  bought  10  :  how  many  did  he  then  have? 

17.  A  farmer  had  35  bushels  of  grain ;  a  part  having 
been  wasted,  he  found  there  were  but  22  bushels  remain- 
ing  :  how  much  was  wasted  ? 

18.  John's  father  is  36  years  old  ;  John  is  12 :  in  how 
paany  years  will  he  be  as  old  as  his  father  now  is  ? 

19.  I  had   65    cents ;    spent  20   cents  for  a  book  and 
ten  for  a  slate :  how  many  had  I  left  ? 


SUBTRACTION.  27 

.  If  you  take  10  from  the  sum  of  two  numbers  there 
be  8  left :  what  is  their  sum  ? 

21.  If  you  take  16  from  the  difference  of  two  numbers 
there  will  remain  12  :  what  is  their  difference  ? 

22.  The  sum  of  two  numbers  is  20  :  what  number  musl 
be  added  to  make  their  sum  30  ? 

23.  The  sum  of  two  numbers  is   16  more -than  thei/ 
difference ;  if  their  difference  is  4,  what  is  their  sum  ? 

24.  The  greater  of  two  numbers  is  12,  and  their  differ- 
ence 5  :  what  is  the  less? 

25.  The  sum  of  two  numbers  is  21 ;  the  less  number  is 
eight :  what  is  the  greater  ? 

26.  If  you  take  one  number  from  another,  what  is  the 
operation  called  ?     Ans.  Subtraction. 

27.  If  you  add  one  number  to  another,  what  is  the 
operation  called?     Ans.  Addition. 

28.  In  what  respect  do  Addition  and  Subtraction  differ? 
Ans.   One  is  exactly  the  reverse  or  opposite  of  the  other. 

29.  When  you  take   one  number  from  another,  what 
do  you  call  that  which  is  left? 

Ans.  Difference  or  Remainder. 

30.  The  remainder  and  less  number,   added  together 
are  always  equal  to  what  ?     Ans.   The  greater  number. 


SECTION   IV.— REVIEW. 
LESSON  I. 

1.  James  had  13  marbles;  he  gave  2  to  Henry,  and 
three  to  Thomas :  how  many  had  he  left  ? 

ANALYSIS. — He  bad  as  many  left  as  the  difference  between 
13  marbles,  and  the  sum  of  2  marbles  and  3  marbles ;  the  sum 
of  2  marbles  and  3  marbles  is  5  marbles;  5  from  13  leaves  8 
marbles ;  therefore,  if  James  had  13  marbles,  and  gave  2  to 
Henry  and  3  to  Thomas,  he  had  8  left. 


28  KAY'S  INTELLECTUAL  ARITHMETIC. 

2.  A   merchant  had  40  barrels  of   flour;  he  sold  to 
one  man  9,  to  another  21 :  how  many  had  2le  left? 

3.  On   Christmas   day,   William    had    36    cents   given 
him;  ho    spent    6    cents   for    apples,    9    cents   for   cakes, 
and  10  cents  for  candy :  how  many  had  he  left? 

4.  A   man  paid   30   dollars  for  a  horse,  the   keeping 
cost  9  dollars,  and  he  sold  him  for  29  dollars :  how  many 
dollars  did  he  lose  ? 

5.  A   man    having    34    dollars,    bought   a   barrel    of 
molasses  for  15  dollars,  and  a  bag  of  coffee  for  10  dol- 
lars :  how  many  dollars  had  he  left  ? 

6.  A  grocer  bought  some  oranges  for  9  dollars,  some 
lemons  for  7  dollars,  some  prunes  for  5  dollars,  and  some 
figs  for  9  dollars,  and   then  sold   them  for  41   dollars: 
how  much  did  he  gain  ? 

7.  A  lady  bought  a  comb  for  25  cents,  some  pins  for 
10  cents,  tape   for  7  cents,  thread   for  6   cents,  and  toy 
books  for  5  cents ;  she  gave  60  cents  to  the  shopkeeper  : 
how  much  change  ougjht  she  to  receive  ? 

a  o 

8.  Two  boys  commenced  playing  marbles ;   each  had 
18  when  they  began ;  when  they  quit,  one  had  25 :  how 
many  had  the  other  ?  * 

9°.  Thomas  has  7  marbles,  David  5,  and  Moses  11; 
how  many  have  they  altogether  ?  How  many  have  Moses 
and  D?vid  together  more  than  Thomas? 

10°.  Three  boys  commence  playing  marbles :  Thomas 
had  20,  David  10,  and  Moses  4;  when  they  quit,  David 
had  6  and  Moses  8 :  how  many  had  Thomas  ? 

11°. A  farmer  had  24  sheep:  9  of  them  were  killed  by 
wolves,  5  of  them  were  stolen,  and  6  he  sold :  how  many 
had  he  left? 

12.  A    grocer  bought  sugar  for  12   dollars,   flour  for 
six   dollars,  and   coffee  for  5  dollars  ;  he   sold  the  whole 
for  30  dollars  :  how  much  did  he  make  ? 

13.  A  lady  had  50  cents ;  she  spent  25  cents  for  butter, 
and  10  cents  for  eggs :  how  much  had  she  left? 


REVIEW.  29 

14.  A  man  is  indebted  to  A,  5  dollars;  to  B,  6  dollars; 
and  to  C,  10  dollars:  he  has  cash  to  the  amount  of  20 
dollars,  and  goods  valued  at  10  dollars  :  should  he  pay 
his  debts,  how  much  would  he  be  worth  ? 

LESSON  II. 

1.  Four  and  3,  less  2,  are  how  many? 

ANALYSIS. — As  many  as  the  difference  between  2,  and  Hit 
sum  of  4  and  3 :  4  and  3  are  7  j  7  less  2  are  5  ;  therefore,  4 
and  3  less  2  are  5. 

2.  Five    and  6  and  2,  less  8,  are  how  many? 
3o  Seven  and  4  and  3,  less  5,  are  how  many  ? 

4.  Eight  and  5  and  4,  less  3,  are  how  many  ? 

5.  Two     and  3  and  5,  less  7,  are  how  many  ? 

6.  Six      and  4  and  3,  less  6,  are  how  many  ? 

7.  Six      and  3  and  5,  less  7,  are  how  many  ? 

8.  Seven  and  4  and  6,  less  5,  are  how  many  ? 

9.  Eight  and  5  and  4  and  6,  less  5,  are  how  many  ? 

10.  Four    and  7  and  6  and  5,  less  8,  are  how  many  ? 

11.  Five    and  8  and  4  and  9,  less  7,  are  how  many  ? 

12.  Seven  and  5  and  8  and  5,  less  6,  are  how  many  ? 

13.  Eight  and  4  and  3  and  9,  less  5,  are  how  many  ? 

14.  Nine    and  5  and  8  and  3  and  1,  less  7,  are  how 
many? 

15.  Eight  and  6  and  5  and  2  and  4  and  3,  less  4,  are 
how  many? 

16.  Seven  and  4  and  8  and  5  and  6  and  2  and  5,  less 
eight,  are  how  many? 

17.  Nine  and  7  and  5  and  3  and  6  and  8  and  7,  less 
six,  are  how  many? 

18.  Eleven  and  4  and  6  and  5  and  7  and  9,  less  3,  are 
how  many  ? 


80  RAY'S   INTELLECTUAL  ARITHMETIC. 

19.  Twelve  and  5  and  7  and  6  and  S  and  3  and  4, 
less  7,  are  how  many  ? 

20.  Twelve  and  7  and  6  and  4  and  5   and  8  and  2, 
less  7,  are  how  many  ? 

21.  Eleven  and  6  and  5  and  3  and  6  and  8  and  6, 
less  9,  are  how  many? 

22.  Eleven  and  7  and  5  and  4  and  5  and  9  and  6, 
less  8,  are  how  many  ? 

23.  Eleven  and  9  and  8  and  7  and  6  and  5  and  4, 
less  3,  are  how  many  ? 

24.  Twelve  and  9  and  7  and  6  and  5  and  3,  less  7, 
are  how  many  ? 

25.  Thirteen  and  4  and  5  and  7  and  8  and  6  and  9, 
less  5,  are  how  many? 

26.  Thirteen  and  7  and  3  and  9  and  8  and  6  and  5, 

less  8,  are  how  many  ? 

27.  Eighteen  and  9  and  10  and  8  and  7,  less  9,  are 
how  many  ? 

28.  Twenty-one  and  5  and  6  and  7  and  8  and  9  and 
10  and  9,  less  8,  are  how  many? 

29.  Seventy,  less  10  and  9  and  8  and  6  and  5  and  4  and 
3  and  2  and  4  and  5  and  6,  are  how  many  ? 

LESSON  III. 

1.  Henry  had  24  cents,  and  spent  all  but  15  :    how 
many  did  he  spend  ? 

2.  A  man  bought  a  cask  of  wine  containing  27  gal- 
lons ;  after  selling  10  gallons  he  found  there  were  but  9 
gallons  remaining,  the  rest  having  leaked  out :  how  much 
did  he  lose  ? 

3.  If  from    20   you   take    12  less  3,  how  many  will 
remain  ? 

4.  If  from  the  sum  of  19  and  10,  you  take  the  differ- 
ence between  17  and  10,  what  will  be  left? 


REVIEW.  3j 

5.  A  man  owed  60  dollars :  lie  paid  at  one  time  20 
dollars,  and  at  another  30  dollars:  he  afterward  borrowed 
5  dollars :  how  much  does  he  still  owe  ? 

6°.  A  maft  paid  38  dollars  for  a  horse,  and  20  for  a 
colt:  he  afterward  sold  the  colt  for  10  dollars,  and  the 
horse  for  65 :  how  much  did  he  make  by  the  transaction  ? 

7.  Twenty-four  less  8,  and  12  less  5,  are  together  how 
much  less  than  25  ? 

8.  Engaged  to  do  a  piece  of  work  for  60  dollars :  had 
an  assistant  25  days  at  a  dollar  a  day,  and  paid  20  dollars 
for  materials  :  how  much  did  I  clear  ? 

9.  If  from  the  sum  of  8  and  9  and  10  and  11,  you 
take  the  sum  of  4  and  5  and  6  and  7,  what  will  you  have 
remaining  ? 

10.  A  jeweler  bought  a  watch  for  40  dollars,  a  chain 
for  15  dollars,  and  a  key  for  3  dollars :  he  sold  them  for 
63  dollars  :  what  did  he  gain  ? 

11.  A  drover  bought  sheep  as  follows  :  of  one  man  10  ; 
of  another,  12 ;  of  another,  5 ;  of  another,  3 :  he  sold  at 
one  time  15  ;  and  at  another,  5  :  how  many  were  left  ? 

12.  A  gentleman  having  40  dollars,  purchased  a  suit  of 
clothes :  his  pants  cost  7  dollars ;  vest,  5  dollars ;  coat,  25 
dollars  :  how  much  had  he  left? 

13.  What  number  must  be  added  to  25,  to  make  a 
gum  14  less  than  45  ? 

14.  What  number  must  be  taken  from  62,  to  give  a 
result  which  shall  be  12  more  than  45  ? 

15.  If  from  the  sum  of  25  and  10  and  12,  you  take  the 
difference  between  28  and  19,  what  will  remain  ? 

16.  A  man  bought  a  horse  for  40  dollars :    and  after 
paying  15  dollars  for  keeping  him,  sold  him  for  75  dol- 
lars :  how  much  did  he  make  ? 

17.  A  gentleman  engaged  in  trade  with   75   dollars : 
after  losing  at  one  time  10  dollars,  and  at  another  5,  he 
gained  20  dollars  :  how  much  did  he  then  have  ? 


32  RAY'S    INTELLECTUAL   ARITHMETIC. 

SECTION  V.— MULTIPLICATION. 

LESSON  I. 

1.  A  boy  gave  2  cents  for  one  lemon,  and  2  cents  foi 
another  :  how  many  cents  did  he  give  for  both  ? 

2.  How  many,  then,  are  2  times  2?     Why?     Because 
2  and  2  are  4. 

3.  A  boy  gave  3  cents  for  one  peach  and  3  cents  for 
another :  how  many  cents  did  he  give  for  both  ? 

4.  How  many,  then,  are  2  times  3?     Why? 

5.  At  4  cents  apiece,  what  will  2  pears  cost? 

6.  How  many  are  2  times  4?     4  times  2?     Why? 

7.  At  3  cents  apiece,  what  will  3  peaches  cost  ? 

8.  How  many  are  3  times  3?     Why? 

9.  At  3  cents  apiece,  what  will  4  apples  cost? 

10.  How  many  are  4  times  3  ?     Why  ? 

11.  At  3  cents  apiece,  what  will  5  pears  cost?     Hov 
many  are  5  times  3?     3  times  5  ? 

12.  At  4  cents  apiece,  what  will  4  lemons  cost?     How 
many  are  4  times  4  ? 

13.  At  5   dollars  a  yard,  what  will  4  yards  of  cloth 
cost?     Htfw  many  are  4  times  5  ?     5  times  4? 

14.  At  6  dollars  a  barrel,  what  will  4  barrels  of  flour 
cost?     How  many  are  4  times  6  ?     6  times  4? 

15.  At  5  cents  apiece,  what  will  5  oranges  cost?     How 
many  are  5  times  5  ? 

16.  At  6  cents  a  yard,  what  will  5  yards  of  tape  cost? 
How  many  are  5  times  6  ?     6  times  5  ? 

17.  At  6  cents  apiece,  what  will  6  oranges  cost?     How 
many  are  6  times  6  ? 

18.  At  7  cents  a  yard,  what  will  2  yards  of  tape  cost? 
How  many  are  2  times  7  ?     7  times  2  ? 


MULTIPLICATION,  33 

19.  At  7  cents  apiece,  what  will  3  lemons  cost?     How 
many  are  3  times  7  ?     7  times  3  ? 

20.  At  7  cents  apiece,  what  will  4  oranges  cost?     How 
many  are  4  times  7  ?     7  times  4  ? 

21.  If  1  marble  is  worth  7  apples,  how  many  apples  are 
worth  5  maizes  ?     How  many  are  5  times  7? 

22.  If  1  peach  is  worth  8  apples,  how  many  apples  are 
worth  2  peaches  ?     How  many  are  2  times  8  ? 

23.  If  1    orange   cost   8   cents,   how    many  cents  will 
three  oranges  cost?     How  many  are  3  times  8? 

24.  If  1  barrel  of  flour  cost  8  dollars,  how  many  dollars 
will  4  barrels  cost?     How  many  are  4  times  8? 

25.  If  1  orange  is  worth   8  apples,  how  many  apples 
are  5  oranges  worth  ?     How  many  are  5  times  8  ? 

26.  At  9  cents  apiece,  what  will  2  oranges  cost?     How 
many  are  2  times  9  ?     9  times  2  ? 

27.  At  9  cents  a  yard,  what. will  3  yards  of  ribbon  cost? 
flow  many  are  3  times  9  ?     9  times  3  ? 

28.  At  10  cents  a  quart,  what  will  3  quarts  of  chest- 
nuts cost?     How  many  are  3  times  10? 

29.  If  1  yard  of  cloth  cost  1 1  dollars,  what  will  3  yards 
cost?     How  many  are  3  times  11? 

30.  At  10  cents  a  bunch,  what  will  4  bunches  of  grapes 
cost?     How  many  are  4  times  10? 

31.  How  many  are  5  times  10?     10  times  5  ? 

SUGGESTION. — The  exercises  given  under  the  subtraction  table 
can  be  extended  to  multiplication.  One  great  use  of  such  exer- 
cises is  to  aw-iken  an  interest  in  the  recitation. 

Take,  for  example,  the  numbers  6  and  6.     Five  and  6  are  11 
6  and  5  are  11 ;  5  from  11  leaves  6 ;   6  from  11  leaves  5;  5  times 
$  are  30;  6  times  5  are  30, 


RAY'S    INTELLECTUAL    ARITHMETIC. 


LESSON   II.— MULTIPLICATION   TABLE. 


1    time     1    is         1   , 

1    time     2     is         2 

1    time     3   is         3 

2    times    1   are      2 

2    times    2     are      4 

2    times    3  are      6 

3    times    1   are      3 

3    times    2     are       6 

3    times    3  are      9 

4   times    1   are      4 

4    times    2     are      8 

4    times    3  are   12 

5    times    1   are      5 

5    times    2     are    10 

5    times    3  are    15 

6    times    1   are       6 

6    times    2     are   12 

6    times    3  are    18 

7    times    1   are      7 

7    times    2     are    14 

7    times    3  are   21 

8    times    1   are      8 

8    times    2     are    16 

8    times    3  are   24 

9    times    1   are      9 

9    times    2     are    18 

9    times    3  are   27 

10    times    1   are    10 

10    times    2     are   20 

10    times    3  are    30 

11    times    1   are    H 

11    times    2     are   22 

11    times    3  are   33 

12    times    1   are    12 

12    times    2     are   24 

12    times    3  are   36 

1    time     4  is        4 

1    time     5    is         5     j        1    time     6   is         6 

2    times    4  are      8 

2    times    5    are    10 

2    times    6   are    12 

3    times   4  are   12 

3    times    5    are    15 

3    times    6   are    18 

4    times   4  are   16 

4    times    5    are    20 

4    times    6  ar&   24 

5    times   4  are  20 

5    times    5    are    25 

5    times    6   are   30 

6    times    4  are   24 

6    times    5    are    30 

6    times    6  are    36 

7    times   4  are  28 

7    times    5    are    35 

7    times    6  are   42 

8    times   4  are  32 

8    times    5    are   40 

8    times    6  are   48 

9    timeg    4  are   36 

9    times    5    are   45 

9    times    6  are   54 

10    times   4  are  40 

10    times    5    are    50 

10    times    6  are    60 

11    times    4  are   44 

11    times   5    are    55 

11    times    6  are    66 

12    times   4  are  48 

12    times    5    are    60 

12    times    6  are   72 

1    time     7   is         7 

1    time     8     is         8 

1    time     9  is          9 

2    times    7  are    14 

2    times    8     are    16 

2    times    9  are    18 

3    times    7   are   21 

3   times    8     are    24 

3    times    9  are     27 

4   times    7  are   28 

4    times    8     are    32 

4    times    9  are    36 

5    times    7  are   35 

5    times    8     are    40 

5    times    9  are    45 

6    times    7  are   42 

6    times    8     are    48 

6    times    9  are    54 

7    times    7  are  49 

7   times    8     aro    56 

7    times    9  are    63 

.  8    times    7  are   56 

8    times    8     are    64 

8    times    9  are    72 

9    times    7  are    63 

9    times    8     are    72 

9    times    9  are    81 

10    times    7   are    70 

10    times    8     are    80 

10    times    9  are     90 

11    times    7  are   77 

11    times    8     are    88 

11    times    9  are    99 

12    times    7  are   84 

12    times    8     are    96 

12    times    9  are  108 

MULTIPLICATION. 


35 


1    time    10  is        10 

1  time    11  is        11 

1  time    12  is       12 

2    times  10  are     20 

2  times  11  are     22 

2  times  12  are     24 

3    times  10  are     30 

3  times  11  are     33 

3  times  12  are     36 

4    times  10  are     40 

4  times  11  are     44 

4  times  12  are    48 

5    times  10  are     50 

5  times  11  are     55 

5  times  12  are     60 

6    times  10  are     60 

6  times  11  are     66 

6  times  12  are    72 

7    times  10  are     70 

7  times  1  1  are     77 

7  times  12  are    84 

8    times  1Q  are     80 

8  times  11  are     88 

8  times  12  are     96 

9    times  10  are     90 

9  times  11  are     99 

9  times  12  are  108 

10    times  10  are  100 

10  times  11  are  HO 

10  times  12  are  120 

11    times  10  are  HO 

11  times  11  are  121 

11  times  12  are  132 

12    times  10  are  120 

12  times  11  are  132 

12  times  12  are  144 

EXERCISES   ON   THE   TABLE. 


7  times    4?      9   times 
7  times  5?     6  times  7? 


How  many  are  4  times   7? 
3?     6  times  5?     8  times  2? 
?  times  8? 

Eight  times  6  ?     6  times  9  ?     9  times  7  ?     8  times  5  ? 

7  times  3?     9  times  8?    7  times  7? 

Nine  times  6  ?  8  times  7  ?  7  times  6  ?  9  times  9  ?  7 
times  9?  10  times  5?  4  times  11?  12  times  3?  8 
times  8  ?  6  times  11  ?  7  times  10  ?  8  times  9  ? 

How  many  are  2  times  2  ?  3  times  2  ?  4  times 
2  ?  3  times  3  ?  5  times  2  ?  4  times  3  ?  7  times  2  ?  5 
times  3  ?  4  times  4  ?  9  times  2  ? 

Five  times  4?  3  times  7?  11  times  2?  8  times  3? 
5  times  5?  9  times  3?  7  times  4?  10  times  3?  8 
times  4?  11  times  3? 

Seven  times  5  ?  9  times  4?  4  times  10?  7  times  6? 
11  times  4?  9  times  5  ?  8  times  6  ? 

Seven  times  7  ?     10  times  5  ?    9  times  6  ?    11  times  5  ? 

8  times  7?  5  times  12?  9  times  7?   8  times  8?    11  times  6? 

Ten  times  7  ?     9  times  8?     11  times  7?     8  times  10? 

9  times  9?    7  times  12?     8  times  11?     9  times  10?    8 
times  12?     11  times  9?     10  times  10  ? 


36  BAY'S   INTELLECTUAL   ARITHMETIC, 

LESSON   III. 

1.  At  2  cents  each,  what  will  7  oranges  cost? 

ANALYSIS. — Seven  oranges  will  cost  7  times  as  much  as  i 
orange.  If  1  orange  cost  2  cents,  7  oranges  will  cost  7  times 
2  cents,  which  are  14  cents;  therefore,  7  oranges  at  2  cents 
each  will  cost  14  cents. 

2.  At  7  cents  each,  what  will  3  melons  cost? 

3.  At  6  cents  a  dozen,  what  cost  5  dozen  apples? 

4.  At  6  cents  a  pound,  what  cost  7  pounds  of  beef? 

NOTE.— The  dollar  sign,  $,  will  now  be  used  in  place  of  the 
word  dollar :  thus,  $-5,  $(3 ;  read  5  dollars,  t>  dollars. 

5.  At  $6  a  pound,  what  cost  8  pounds  of  opium? 

6.  At  $3  a  barrel,  what  cost  9  barrels  of  cider? 

7.  At  $4  a  pair,  what  cost  7  pairs  of  boots  ? 

8.  At  8  cents  a  dozen,  what  cost  10  dozen  pens? 

9.  What  cost  6  yards  of  cloth  at  $7  a  yard  ? 

10.  What  cost  8  barrels  of  flour  at  $5  a  barrel? 

11.  If  a  man  travel  7  miles  an  hour,  how  far  will  lie 
travel  in  8  hours  ? 

12.  On   a   chessboard    are   8  rows   of   squares,  and  8 
squares  in  each  row:  how  many  squares  on  the  board? 

13.  An  orchard  has  11  rows  of  trees,  and  7  trees  in 
each  row :  how  many  trees  in  the  orchard? 

14.  In  1  cent  are  10  milis ;  how  many  mills  are  there 
in  3  cents?     In  4?     In  5?     6?     7?     8? 

15.  In  1  pint  are  4  gills ;  how  many  gills  are  there  in 
2  pints?     In  3?     In  4?     5?     6?     7?     8? 

16.  In    1    bushel   are   4   pecks ;    how  many   pecks   are 
there  in  2  bushels?     In  3?     In  4?     5?     6?     7  ? 

17.  In  1  peck   are  8  quarts ;     how   many  quarts   ara 
&ere  in  2  pecks ?     In  3?     In  4?     5?     6?     7? 


MULTIPLICATION.  37 

18.  In  1  bushel  how  many  quarts?     Why  ? 

19.  What  will  9  yards  of  cloth  cost  at  $6  a  yard? 

20.  What  will  9  oranges  cost  at  8  cents  each? 

21.  Two  men  start  from  the  same  place  and  travel  in 
apposite  directions :   one  travels  2  miles  an  hour,  the  other 
4  miles :  how  far  will  they  be  apart  at  the  end  of  1  hour? 
At  the  end  of  2  hours  ?     3  hours  ? 

22.  If  2   men   can   do  a  job  of  work  in  3  days,  how 
many  days  will  it  take  1  man  to  do  it  ? 

ANALYSIS. — It  will  require  1  man  twice  as  long  as  2  men. 
If  it  take  2  men  3  days,  it  will  take  1  man  twice  3  days^ 
which  are  6  days;  therefore,  if  2  men  do  a  job  of  work  in 
3  days,  1  man  will  do  it  in  6  days. 

23.  If  3  men  can  do  a  piece  of  work  in  4  days,  in  how 
many  days  can  1  man  do  it  ? 

24.  If  4  men  can  do  a  piece  of  work  in  6  days,  in  how 
many  days  can  1  man  do  it? 

25.  If  a  quantity  of  bread  serve  8  men  4  days,  how 
many  days  will  it  serve  1  man  ? 

26.  If  a  man  can  earn  $6  in  1  week,  how  many  dol- 
lars can  he  earn  in  8  weeks  ? 

27.  A  person  has  a  job  of  work  which  6  men  can  ao 
in  9  days ;  but  it  is  necessary  to  do  it  in  one  day :  how 
many  men  must  be  employed  ? 

28.  If  2  barrels  of  cider  last  6  persons  4  weeks,  how 
many  weeks  will  it  last  1  person  ? 

29.  If  $9  worth  of  provisions  last  8  persons  11  days, 
how  many  person^  will  it  last  1  day  ? 

30.  If  the  interest  of  $1  is  6  cents  a  year,  what  will  be 
the  interest  for  2  years?      For  3  .years?      For  4?      5? 
For  6?     7?     8?     9?     10? 

31°.  I  bought  6  barrels  of  apples  at  $2  a  barrel,  and  4 
barrels  of  sugar  at  $11  a  barrel :  how  much  did  they 
both  cost? 

2d  Bk.  3 


38  HAY'S   INTELLECTUAL   ARITHMETIC. 

32.  Four  and  4  are  8,  and  4  are  12,  and  4  are  16,  and 
4  are  20 :  here  we  find  that  4  taken  five  times  makes  20. 
What  is  this  operation  called?     Ans.  Addition. 

33.  When  we  say  5  times  4  make  20,  what  do  we  call 
the  operation?     Ans.  Multiplication. 

34.  How,  then,  would  you  define  Multiplication  ?     Ans. 
Multiplication  is  a  short  method  of  performing  several  ad- 
ditions of  the  same  number. 

35.  The  number  to  be  multiplied,  4,  is  called  the  mul- 
tiplicand ;  the  number  you  multiply  by,  5,  the  multiplier; 
and  the  answer,  20,  the  product. 


LESSON  IV. 

1.  Bought  2  apples  at  2  cents  each,  2  pears  at  3  cents 
each,  and  an  orange  for  5  cents :  what  did  they  cost  ? 

2c  Two  men  start  from  the  same  place  and  travel  in 
the  same  direction ;  one,  5  miles  an  hour  ;  the  other,  7 
miles :  how  far  will  they  be  apart  in  10  hours  ? 

3.  If,  in  the  above  question,  the  men  travel  in  opposite 
directions,  how  far  will  they  be  apart  in  12  hours? 

4.  A  lady  went  shopping  with  $15;  she  bought  4  yards 
of  cloth  at  82  a   yard ;  2  pairs   of  gloves  at  $1  a  pair ; 
and  a  shawl  for  $2 :  what  did  they  cost,  and  how  much 
had  she  left? 

5.  A  man  bought  4  peaches  at  5  cents  each,  3  pears 
at  3  cents  each,  and  2  pints  of  chestnuts  at  5  cents  a 
pint :  how  much  did  they  cost  ? 

6.  What  will  be  the  sum  of  3  and  9  and  7?  less  the 
sum  of  8  and  6  and  1  ? 

7.  If  a  man  earn  5  shillings  a  day,  and  a  boy  3  shil- 
lings, how  much  will  both  earn  in  7  days  ? 

8.  A  drover  gave  $10  and  7  sheep,  valued  at  $4  * 
head,  for  a  cow  and  calf:  how  much  did  they  cost? 


MULTIPLICATION.  39 

9°.  A  merchant  sold  cloth  at  $7  a  yard :  a  tailo*  bought 
of  this  cloth,  at  one  time,  5  yards,  and  at  another,  3  yards: 
what  was  the  amount  of  his  bill  ? 

10  .Two  brothers,  Henry  and  Rufus,  each  received  for 
their  work  3  dimes  a  day :  how  much  did  they  both  re- 
ceive  for  6  days'  work  ? 

11.  If   12   horses  can   be   sustained   in   a   pasture  10 
months,  how  many  horses  will  it  feed  1  month? 

12.  What  is  3  times  the  difference  between  15,  and  tho 
sum  of  5  and  2  ? 

13.  The  sum  of  two  numbers  is  23;  the  smaller  is  11 : 
what  is  5  times  the  larger? 

14.  The   difference  between  two  numbers  is  7 :  if  the 
larger  be  12,  what  will  8  times  the  smaller  be  ? 

15.  If  a  boy  buy  apples  at  1  cent  each,  and  sell  them 
for  3  cents  each,  what  would  he  make  if  he  purchase  10 
cents'  worth  of  apples? 

16.  George  bought   a  book   for   50  cents  and  sold  it 
for  $1 :  what  would  he  have  made,  had  he  bought  2  books, 
and  sold  them  at  the  same  rate  as  the  first  ? 

17  .Albert  has  5  times  two  marbles  less  than  50,  and 
Edward  has  5  times  two  more  than  50  :  how  many  has 
each  ? 

18.  If  $1  gain  $3  in  a  year,  what  will  $12  gain  in 
double  the  time  ?' 

19.  A  man  bought  a  cask  of  wine  containing  20  gallons, 
at  $1  a  gal. ;  5  gal.  having  leaked  out,  he  sold  the  remain- 
der  at  $2  a  gal. :  how  much  did  he  make  ? 

20.  Two  men  start  from  the  same  place,  at  the  same 
time,  and  tiavel  the  same  way ;  if  they  travel  at  the  same 
rate,  how  far  will  they  be  apart  at  the  end  of  10  hours? 

If  one  goes  10  miles  an  hour,  and  the  other  7,  how  fai 
will  they  be  apart  in  7  hours  ? 

If  they  go  in  opposite  directions,  each  at  the  rate  of  5 
miles  per  hour,  how  far  will  they  be  apart  in  9  hours  ? 


40  RAY'S    INTELLECTUAL   ARITHMETIC, 

21.  A  stage  starts  from  a  certain  town,  and  travels  at 
the  rate  of  8  miles  per  hour :  at  the  same  time,  another 
starts  from  the  same  place,  and  travels  in  the  same  direc- 
tion, 4  miles  per  hour :  how  far  will  they  be  apart  at  the 
end  of  12  hours  ? 

22.  A  grocer  bought   10  pounds  ot  tea  at  7  shillings 
a  pound ;    after  using  3   pounds,  he  sold  the  remainder 
at  10   shillings  a  pound :    how  much  did   the  3  pounds 
which  he  used  cost  him,  in  the  end  ? 

23.  Bought   6   bushels  of   corn  at  5  dimes  a  bushel ; 
sold  4  bushels  at  6  dimes  a  bushel,  and  2  bushels  at  4 
dimes  a  bushel :  how  much  did  I  make  ? 

24.  If  an  orange  cost  5  cents,  and  an  apple  2  cents, 
what  will  2  oranges  and  4  apples  cost  ? 

25.  If  pork  is  8  cents,  and  beef  10  cents  a  pound,  what 
cost  7  pounds  of  pork  and  6  pounds  of  beef? 

26.  If  an  orange  cost  5  times  as  much  as  an  apple,  how 
much  more  will  6  oranges  cost  than  25  apples,  if  an  apple 
is  worth  1  cent? 

27.  If  a  pound  of  sugar  cost  5  cents,  and  a  pound  of 
coffee  3  times  as  much,  less  3  cents,  what  will  be  the  cost 
of  3  pounds  of  sugar  and  2  of  coffee  ? 

*28  .Bought,  at  one  time,  5  yards  of  muslin  at  10  cents 
a  yard ;  at  another,  10  yards  at  5  cents  a  yard :  how 
much  did  it  all  cost? 

29.  When  salt  is  4  cents  a  quart,  and  molasses  3  times 
as   much  lacking  2  cents,  what  would  be  the  cost  of   3 
quarts  of  molasses  and  2  of  salt  ? 

30.  If  a  man  earn  $15   per   week,   and   spend  $11    a 
week,  how  much  will  lae  save  in  3  weeks?     How  much 
can  he  save  in  two  months  of  4  weeks  each  ? 

*  In  the  present  edition  of  this  volume,  a  very  few  examples 
have  been  modified,  the  better  to  adapt  them  to  a  more  perfect 
gradation.  Such  are  indicated  by  the  character,  0.  The  omission 
of  these  examples  in  classes  having  books  of  the  present  and  for- 
Bier  edition,  will  obviate  all  confusion. 


DIVISION,  41 


SECTION  VI.— DIVISION 

LESSON  I. 

1.  At  1  cent  each,  how  many  cakes  can  you  buy  fot 
4  cents  ?     1  in  4  how  many  times  ?     Why  ?     Ans.  Be- 
cause 4  times  1  are  4. 

2.  At  2  cents  each,  how  many  apples  can  you  buy 
for  4  cents  ? 

ANALYSIS. —  You  can  buy  as  many  apples  as  2  cents  are 
contained  times  in  4  cents ;  2  cents  are  contained  in  4  cents 
2  times ;  therefore,  at  2  cents  each,  you  can  buy  2  apples  for 
4  cents. 

Two  in  4  how  many  times  ?  Why  ?  Because  2  times 
2  cure  4. 

3.  Among  how  many  boys  can  6  apples  be  divided, 
giving  to  each  boy  two  apples  ?     2  in  6  how  many  times  ? 
Why  ?     Because  3  times  2  are  6. 

4.  At  2  cents   each,    how  many  apples  can  you  buy 
for  8  cents  ?     2  in  8  how  many  times  ?     Why  ? 

5.  At  3  cents  each,  how  many  peaches  can  you  buy 
for  6  cents  ?     3  in  6  how  many  times  ?     Why  ? 

6.  At   3    cents   each,    how   many  pears  can  you  buy 
for  9  cents  ?     3  in  9  how  many  times  ?     Why  ? 

7.  At  2  cents  each,  how  many  cakes  can  you    buy 
for  10  cents  ?     2  in  10  how  many  times  ?     Why  ? 

8.  At  2  cents  each,  how  many  marbles  can  you  buy 
for  14  cents  ?     2  in  14  how  many  times  ?     Why  ? 

9.  At  5  cents  each,  how  many  lemons  can  you  buy 
for  15  cents  ?     5  in  15  how  many  times  ?     Why? 

10.  A  boy  has  16  marbles,  and  wishes  to  divide  them 
into  piles  of  2  each :  how  many  piles  will  there  be  1 
How  many  twos  in  16  ?  Why  ? 


42  RAY'S    INTELLECTUAL   ARITHMETIC. 

11.  At  3  cents  each,  how  many  peaches  can  you  buy 
for  18  cents  ?     3  in  18  how  many  times  ? 

12.  At  5  cents  each,  how  many  oranges  can  you  buy 
for  20  cents  ?     5  in  20  how  many  times  ? 

13.  At  $3  a  yard,  bow  many  yards  of  cloth  can  yov 
buy  for  $21  ?     3  in  21  how  many  times  ? 

NOTE. — The  dollar  sign,  $,  is  used  ie.  place  of  the  word  dollar  ; 
thus,  $4,  $7 ;  read  4  dollars,  7  dollars. 

14.  A  lady  spent  22  cents  for  tape,  at  2  cents  a  yard: 
how  many  yards  did  she  buy  ? 

15.  At  6  cents  each,  how  many  oranges  can  you  buy 
for  24  cents  ?     How  many  at  8  cents  each  ?     6  in  24  how 
many  times  ?     8  in  24  how  many  times  ? 

16.  In  an  orchard  of  25  apple  trees  there  are  5  rows; 
how  many  trees  in  each  row  ? 

ANALYSIS. —  One  tree  in  each  row  requires  5  treec ;  hence, 
there  will  be  as  many  rows,  as  5  trees  are  contained  times  in 
25  trees ;  5  trees  in  25  trees,  5  times.  Ans.  5  rows. 

17.  If  a  man  can  travel  3  miles  in  an  hour,  how  many 
hours  will  it  take  him  to  travel  27  miles  ?     3  in  27  how 
many  times  ? 

18.  A  man  gave  §28  for  sheep,  at  $4  a  head :  how  many 
did  he  buy  ?     4  in  28  how  many  times  ? 

19.  If  you  had  30  cents,  how  many  marbles  could  you 
buy  at  3  cents  each  ?     3  in  30  how  many  times  ? 

20.  There  are  32  cents  on  a  table,  in  4  piles  :  how  many 
in  eacn  pile?     4  in  32  how  many  times? 

21.  In  an  orchard  containing  35  apple  trees,  there  are 
5  rows :  how  many  trees  are  there  in  each  row  ?     5  in  35 
how  many  times  ? 

22.  Six   men  receive   $36  for  a  job  of  work:  what  is 
each  man's  share  ?     6  in  36  how  many  times  ? 


DIVISION. 


43 


23.  Four  quarts  make  1  gallon  :  how  many  gallons  in 
36  quarts  ?     4  in  36  how  many  times  ? 

24.  If  a  man  travel  10  miles  in  1  hour,  in  how  many 
hours  will  he  travel  40  miles  ? 

25.  Forty-two  cents  were  diyided  equally  among  6  boys; 
how  many  cents  did  each  boy  receive  ? 

26.  Forty-two  are  how  many  times  7  ? 

27.  If  you  divide  45  apples  equally  among  9  boys,  how 
many  apples  will  each  boy  receive  ? 


LESSON  II— -DIVISION  TABLE. 


2    in       2       1    timo 

3    in       3       1   .time 

4    in      4       1    time 

2    in       4       2    times 

3    in       6       2    times 

4    in      8       2    times 

2    in       6       3    times 

3    in       9       3    times 

4    in    12       3    times 

2    in       8       4    times 

3    in    12       4    times 

4    in    16       4    times 

2    in    10       5    times 

3    in    15       5    times 

4    in    20       5    times 

2    in    12       6    times 

3    in    18       6    times 

4    in    24       6    times 

2    in    14       7    times 

3    in    21       7    times 

4    in    28       7    times 

2    in    16       8    times 

3    in    24       8    times 

4    in    32       8    times 

2    in    18       9    times 

3    in'  27       9    times 

4    in    36       9    times 

2    in    20    10   times 

3    in    30    10    times 

4    in    40    10    times 

2    in    22    11    times 

3    in    33    11    times 

4    in    44  '11    times 

2    in    24    12    times 

3    in    36    12    times 

4    in    48    12    times 

5    in       5        1    timo 

6    in       6       1    time 

7    in       7       1     time 

5    in    10        2    times 

6    in    12       2    times 

7    in    14       2    times 

5    in    15       3    times 

6    in    18       3    times 

7    in    21       3    times 

5    in    20       4    times 

6    in    24       4   times 

7    in    28       4    times 

5    in    25       5    times 

6    in    30       5    times 

7    in    35       5    times 

5    in   30       6    times 

6    in    36       6    times 

7    in   42       6    times 

5    in    35       7    times 

6    in   42       7    times 

7    in    49       7    times 

5    in    40       8    times 

6    in    48       8    times 

7    in    56       8    times 

5    in    45        9"  times 

6    in    54       9    times 

7    in    63       9    times 

5    in    50     10    times 

6    in    60    10    times 

7    in    70    10    times 

5    in    55     11    times* 

6    m    66    11    times 

7    in    77     11     times 

5    in    60     12    times 

6    in    72    12    times 

7    in    84    12    timeq 

44     . 


BAY'S   INTELLECTUAL  ARITHMETIC. 


8  in      8      1  time 

9  in          91  time 

10   in       10      1     time 

8  in    16      2  times 

9   in       18      2  times 

10  in       20      2    times 

8  in    24      3  times 

9  in       27      3  times 

10  in       30      3    timea 

8  in    32      4  times 

9  in       36      4  times 

10  in       40      4    times 

8  ia   40      5  times 

9  in       45      5  times 

10  in       50      5    times 

8  in    48      6  times 

9   in       54      6  times 

10   in       60      6    times 

8  in    56      7  times 

9  in       63      7  times 

10   in       70      7    times 

8  in    64      8  times 

9  in       72      8  times 

10   in       80      8    times 

8  in    72       9  times 

9  in        81       9  times 

10   in       90      9     times 

8  in    80    10  times 

9  in       90   10  times 

10  m    100   10    times 

8  in    88    11  times 

9  in       99    11  times 

10  in    110   11    times 

8  in    96    12  times 

9  in    108   12  times 

10   in    120   12     times 

11   in    11      1  time 

11   in       55        5  times       11    in       99       9      times 

11  in   22      2  times 

11     n       66        6  times 

11   in    110    10     times 

11  in   33      3  times 

11     n      77       7  timea 

11   in    121    11      timos 

11  in   44      4  times 

11     n      88       8  times 

11   in    132    12     times 

12  in    12      1  time 

12    n       60      5  times 

12   in    108       9     times 

12  in    24      2  times 

12    n      72      6  times 

12   in    120    10     timea 

12  in    36      3  times 

12     n       84      7  times 

12   in    132    11      times 

12  in    48      4  times 

12  in       96      8  times 

12   in    144    12     tiroes 

NOTE.  —  The  four  simple  operations  of  Arithmetic  may  now  be 

combined  in  a  single  example  :  thus, 

10  and      5  are       15                10  times     5  are  50 

5  and    10  are       15                  5  times  10  are  50 

10  from  15  leaves    5                10  in  50     5  times 

5  from  15  leaves  10                  5  in  50  10  times 

LESSON   III. 

1.  Two  in  12  how  many  times?     2  in  16?     2  in  24? 
Sin  9?     3  in  15?     3  in  21  ?     3  in  27  ?     4  in  8  ?    4  in 
20  ?     4  in  28  ?     4  in  36  ?     4  in  48  ? 

2.  Five  in  15  how  many  times  ?     5  -in  30  ?     5  in  45  ? 
5  in  60  ?     6  in  18  ?     6  in  24  ?     6  in  36  ?     6  in  42  ?     6 
in  54  ?     6  in  66  ? 


DIVISION.  45 

3.  Seven  in  14  how  many  times  ?     7  in  28  ?  7  in 
42?     7  in  56?     7  in  63  ?     7  in  84?     8  in  24?  8  in 
40  ?     8  in  56  ?     8  in  72  ?     8  in  96  ? 

4.  Nine  in  18  how  many  times  ?       9  in  27  ?  9  in 
45?     9  in  54?     9  in  63  ?     9  in  81  ?     9  in  108  ?  10  in 
20?     10  in  60?     10  in  90  ?     10  in  100  ? 

5.  Eleven  in  55  how  many  times?      11  in  77  ?     11 
in  99?     11  in  110?     11  in  121  ?     12  in  24  ?     12  in  48  ? 
12  in  60?     12  in  72?     12  in  96?     12  in  108?     12  in 
120  ?     12  in  144  ? 

6.  If  12  peaches  be  divided  equally  among  3  children, 
how  many  will  each  have  ? 

7.  Four  boys  gave  their  sister  24  apples,  each  an  equal 
number  :  how  many  did  each  give  ? 

8.  A  mother  divided  20  cents  equally  between  her  2 
little  girls :  how  many  did  each  receive  ? 

9.  Five    books    cost    35    cents:    how    much  is    that 
apiece  ? 

10.  A  man  has  $40 :  if  he  spend  $5  a  week,  how  long 
will  it  last  ? 

11.  There  are  3  feet  in  1  yard:  how  manv  yards  are 
there  in  21  feet?     In  27  feet?     In  36  feet? 

12.  Four  quarts  make  1  gallon :  how  many  gallons  in 
28  quarts  ?     In  16  ?     In  32  ?     In  36  ?     44  ?     48  ? 

13.  If  5  apples  are  worth  1  pear,  how  many  pears  are 
worth  25  apples  ?     35  apples  ?     45  apples  ? 

14.  If  6  pears  are  worth  an  orange,  how  many  oranges 
can  you  get  for  30  pears  ?     For  42  pears?     For  54  pears? 
For  66  pears? 

15.  If  1  man  do  a  piece  of  work  in  42  days,  how  many 
days  will  it  take  7  men  ? 

16.  If  1  man  eat  a  certain  quantity  of  provisions  in  56 
days,  how  many  days  will  it  last  7  men  ? 

17.  If  1  pipe  empty  a  cistern  in  63  hours,  in  how  many 
hours  will  9  pipes  of  the  same  size  empty  it  ? 


46  BAY'S   INTELLECTUAL    ARITHMETIC. 

18.  Eight  quarts  make  a  peck :  how  many  pecks  in  24 
quarts  ?     In  40  ?     In  56  ?     In  72  ? 

19.  If  hay  is  worth  S9  a  ton,  how  many  tons  can  be 
bought  for  §27  ?     For  $45  ?     For  §54  ?     For  §63  ? 

20.  Ten  men  bought  a  horse  for  §60 :  how  much  did 
each  one  pay? 

21.  If  11  ounces  cost  88  cents:  what  cost  1  ounce? 

22.  A  man  paid  §108  for  12  Saxony  sheep  :  how  inucJ* 
was  that  apiece  ? 

23.  In  an  orchard  there  are  120  trees  in  10  rows :  how 
many  trees  in  each  row  ? 

24.  A  man  earns  §144  in  12  weeks :  how  much  is  that 
a  week  ?     How  much  a  day,  allowing  6  working  days  to 
the  week  ? 

25.  If  6  men  earn  §84  in  7  days,  how  much  do  they 
all  earn  in  1  day? 

26.  If  9  men  earn  §108  in  3  days,  how  much  does  i 
man  earn  ?     How  much  does  each  man  earn  in  a  day  ? 

27.  If  2  from  6  leaves  4,  2  from  4  leaves  2,  and  2  from 
2  leaves  0,  how  many  times  is  2  taken  from  6  ?     Ans.  3. 
What  do  you  call  the  operation  ?     Subtraction. 

28.  Two  is  contained  in  6,  3  times ;  what  do  you  call 
the  operation?     Ans.  Division. 

29.  How  would   you   define   Division?     Ans.  Division 
is  subtracting  the  same  number  several  times;  or,  finding 
koto  many  times  one  number  is  contained  in  another. 

30.  The  number  to  be  divided,  6,  is  called  the  dividend; 
the  number  you  divide  by,  2,  the  divisor ;  and  the  answer, 
8,  the  quotient. 

LESSON   IV. 

1.  Twelve  are  how  many  times  2?     3?     4?     6? 

2.  Twenty -four  are  how  many  times  3?    6?    8?    12? 

3.  Seventy-two  are  how  many  times  12?   8?    0?    9? 


DIVISION.  47 

4.  How  many  oranges  at  5  cents  each,  must  be  given 
for  10  pears  at  2  cents  each  ? 

5.  A  wheel  is  10  feet  in  circumference ;  what  distance 
Will  it  move  in  making  one  revolution  ?    how  many  revolu- 
tions will  it  make  in  going  120  feet? 

6.  An  orchard  contains  10  rows  of  trees,  and  7  trees 
in  a  row ;  if  there  were  but  5  rows,  how  many  trees  would 
there  be  in  a  row  ? 

7.  I  have  three  times   as   many  marbles   as  the  sum 
of   1,  2,   and   3,   is   contained  times  in   60 :    how   many 
have  I  ? 

8.  Bought  6  hats  at  $5  apiece,  and  4  yards  of  cloth 
at  $3  a  yard ;  gave  in  exchange  flour  at  $6  a  barrel :  how 
many  barrels  did  it  take  ? 

9.  If  a  man  gain  6  miles  in  3  hours,  how  long  will 
it  take  to  gain  24  miles  ? 

10.  Two  times  6  are  contained  how  many  times  in  the 
sum  of  36  and  12  ? 

11.  If  60  be  divided  by  some  number,  the  result  will 
be  10  :  what  is  that  number  ? 

12.  I  have  a  number  in  my  mind  which,  divided  by  3, 
gives  2  times  6  :  what  is  the  number  ? 

13  .If  I  purchase  lemons  at  the  rate  of  2  for  6  cents, 
and  sell  7  for  28  cents,  how  much  do  I  gain  ? 

14.  A  man  has  a  job  of  work  which  9  men  can  per- 
form in  2  days ;  he  desires  to  complete  it  in  3  days :  how 
many  men  must  he  employ  ? 

15.  Five  times  the  sum  of  two  numbers  is  equal  to  60; 
if  7  is  one  of  them,  what  is  the  other? 

16.  Henry  has  6  dimes ;  Thomas  twice  as  many  less  2; 
and  Samuel  3  times  as  many  as  Henry :  how  many  have 
they  together  ? 

17  .If  to  the  number  of  times  4  is  contained  in  12,  you 
add  3?  and  subtract  the  result  from  9,  what  will  remain  ? 


48  RAY'S    INTELLECTUAL   ARITHMETIC. 

18.  Five  oranges  were  sold  for  25  cents,  and  10  cents 
were  gained  :  what  did  each  cost  ? 

19.  What  number  subtracted  from  17,  will  leave  double 
the  remainder  that  5  from  9  leaves  ? 

20.  A   boy    said    that  10  taken   from   the   number   of 
apples  he  had,  left  twice  as  great  a  remainder  as  the  dif- 
ference between  1  dozen  and  8  :  how  many  had  he  ? 

21.  A    certain    number    multiplied    by    10,    is    5    less 
than  45  :  what  is  that  number  ? 

22.  If  you  multiply    any   number,    10,   by   any  other 
number,   5,   and   divide  the   product  by  the    same  num- 
ber, 5,  what  will  be  the  result  ? 

23.  If  2  oranges  are  worth  5  apples,  how  many  apples 
are  worth  12  oranges  ? 

ANALYSIS. — As  many  times  5  apples,  as  2  oranges  are  con* 
tamed  times  in  12  oranges. 

24.  One   man   goes    10    miles   while    another  goes  7 ; 
when    the    first   has    gone    90    miles,    how   far   will    the 
second  have  gone  ? 

25°.  James  earns  8  cents  while  John  earns  12;  when 
John  has  earned  60,  how  many  has  James  earned  ? 

26  .George   learns  5    lessons    while  Charles  learns   4: 
how  many  lessons  will  both  have   learned  when  Charle? 
has  learned  20  ? 

27  .  A   man   can   earn    §9   while  a  boy  earns  5  :  how 
many  dollars  will   both   have  earned  when  the  man  has 
earned  $36  ? 

28.  How  many  times  will    the   hammer  of  a  common 
clock  strike,  from  noon  till  6  o'clock  in  the  evening? 

29.  Two    numbers   added    together    make    30 ;    if    the 
greater  number  was  5   more,  and  the  less  3  more,  what 
would  their  sum  then  be  ? 

30.  William  can  count  11  while  James  counts  7:  low 
many  will  James  count  while  William  is  counting  77  ? 


PARTS    OF   NUMBERS.  49 

31.  When   flour   was   5  cents   a   pound  and   sugar   11 
cents,   10  pounds   of  flour  and   5  pounds   of  sugar  were 
given  for  a  box  of  eggs  at  5  cents  a  dozen :  how  many 
dozen  were  in  the  box  ? 

32.  The  sum  of  two  numbers  is  24 ;  if  they  were  both 
equal  to  the  greater,  their  sum  would  be  28 :    what  are 
the  numbers? 


SEC.    VII.— PARTS   OF  NUMBERS. 

LESSON  I. 

If  a  unit  or  whole  thing  is  divided  into  TWO  equal  parts, 
one  of  the  parts  is  called  ONE-HALF. 

1.  When  an  apple  is  divided  into  two  equal  parts,  what 
is  one  part  called?     Ans.   One-half  of  an  apple. 

-    2.  How  many  halves  in  1  apple?-    Ans.  Two  halves. 

3.  How    many   halves   in    2  apples  ?     In   3  ?     In  4  ? 
[n5?     In  6?     In  7?     In  8?     In  9?     In  10? 

ANALYSIS. — In  2  apples  there  are  two  times  as  many  halves 
as  there  are  in  1  apple;  but  in  1  apple  there  are  2  halves,  and 
in  2  apples  there  will  be  2  times  2  halves,  or  4  halves ;  there- 
fore, there  are  4  halves  in  2  apples. 

4.  How  many  halves  in  2  apples  and  one-half  of  an 
apple  ?     In  3  apples  and  one-half?     In  4  and  one-half? 

5.  What  do  you  mean  by  one -half  of  anything? 

If  any  thing  or  any  number  is  divided  into  THREE 
equal  parts,  one  of  the  parts  is  called  ONE-THIRD  of  the 
thing  or  number ;  two  parts  are  called  TWO-THIRDS  •  and 
three  parts,  THREE-THIRDS,  or  the  whole. 

6.  When  an  apple,  or  any  thing  is  divided  into  three 
equal  parts,  what  is  one  part  called  ?     What  two  parts  ? 

7.  How  many  thirds  in  1  apple  ?     In  2  ?     In  3  ?     In 
4?    In  5?     In  6?    In  7?     In  8?     In  9? 


50  RAY'S   INTELLECTUAL   ARITHMETIC. 

8.  How  many  thirds  in  1  apple  and  one-third  of  ** 
apple  ?     In  one  apple  and  two-thirds  of  an  apple  ? 

9.  How   many  thirds   in   two   apples   and    one-third? 
In  2  apples  and  two-thirds  ? 

10.  What    do    you    understand   by    one -third    of    any 
thing?     By  two-thirds?     Ans.    That  a   whole   thing  has 
been  divided  into  3  equal  parts  and  2  of  those  parts  taken. 

If  any  thing  is  divided  into  4  equal  parts,  1  of  the  parts 
is  called  ONE-FOURTH  ;  two  parts  are  called  TWO-FOURTHS, 
and  so  on. 

When  any  thing  is  divided  into  5  equal  parts,  1  of  the 
parts  is  called  ONE-FIFTH;  two  parts,  TWO-FIFTHS;  three 
.parts,  THREE-FIFTHS,  and  so  on. 

11.  When  an  apple  is  divided  into  four  equal  parts,  what 
is  one  part  called  ?     What  two  parts  ?     Ans.  Two-fourths. 
What  three  parts  ?     Am.    Three- fourths. 

12.  How  many  fourths  in  1  apple  ?     In  2  apples  ?     In 
3  apples?     In  4?     In  5  ?     In  6  ?     In  7? 

13.  How  many  fourths  in  one   apple  and  one-fourili? 
In  1  and  two -fourths  ?     In  1  and  three-fourths  ? 

14.  In  2   apples    and    one-fourth    how  many  fourths? 
In  2  apples  and  two-fourths  ?     In  2  and  three -four  ths  ? 

15.  What  are  fourths  often  called  ?     Ans.    Quarter*. 

16.  What   do   you    understand  by   one-fourth   of  any 
thing  ?     By  two -four  ths  ?     By  three-fourths  ? 

17.  When  an  apple  is  divided  into  5  equal  parts,  what 
is  one  part  called?     What  are  2  parts  called?     What  3 
parts?     What  4  parts?     Ans.  Four-fifths. 

18.  How    many   fifths    in   1    apple?      In   2  ?      In    3  ? 
In  4  ?     In  5  ?     In  6  ?     In  7  ?     In  8  ?     In  9  ? 

19.  How   many   fifths    in    1    apple    and    1 -fifth   of  an 
apple  ?   In  1  and  2-fifths?    1  and  3-fifths?    1  and  4-fifths? 

20.  How  many  fifths  in  2  apples  and  1 -fifth  ?     In  2  and 
2-fifths  ?     2  and  3-fifths  ?     2  and  4-fifths  ? 


PARTS   OF   NUMBERS.  51 

21.  What  do  you  understand  by  1 -fifth  of  any  thing? 
By  2-fifths  ?     By  3-fifths  ? 

22.  When  an  apple  is  divided  into  6  equal  parts,  what 
is  1  part  called  ?    What  are  2  parts  called  ?     What  3  parts  ? 
W^hat  4  parts  ?     What  5  parts  ? 

23.  How  many  sixths  in  1  apple  ?     In  2  apples  ?     In 
3  apples?     In  4?     In  5  ?     In  6?     In  7?     In  8? 

24.  How    many  sevenths  in   1   apple  ?     In   2  apples  ? 
Ln3?     In  4?     In  5?     In  6?     In  7?     In  8  ? 

25.  How  many  eighths  in  1  apple?      In   2?      In  3  ? 
In  4?     In  5?     In  6?     In  7?     In  8  ?     In  9? 

26.  How   many  ninths  in   1   apple?      In  2  ?      In  3  ? 
In  4?     In  5?     In  6?     In  7  ?     In  8?     In  9? 

LESSON    II. 

1.  If  a  yard  of  tape  is  worth  2  cents,  what  is  one- 
half  of  it  worth  ?     Ans.  One  cent. 

2.  What   is   one -half  of   2   cents  ?      Ans.     One  cent. 
Why  ?     Ans.  Because  if  you  divide  2  cents  into  two  equal 
parts,  one  of  the  parts  is  1  cent. 

3.  One  is  what  part  of  2  ?    Ans.  One  is  the  half  of  2. 

4.  If  you  can  buy  an  apple  for  2   cents,  how  many 
can  you  buy  for  3  cents?     Ans.   One  apple  and  l-half. 

5.  Three  are  how  many  times  2  ?     Ans.    Once  2  and 
l-half  of  2. 

6.  Four  are  how  many  times  2  ? 

7.  If   2   cents  will  buy  1  yard  of   tape,  how  many 
yards  will  5  cents  buy  ? 

8.  Five  are  how  many  times  2?     -4ns.   Two  times  2 
and  l-half  of  2. 

9.  6  are  how  many  times  2  ?        How  many  2's  ? 

10.  7  are  how  many  times  2  ?        How  many  2's  ? 

11.  8  are  how  many  times  2?       How  many  2's? 


52  RAY'S   INTELLECTUAL  ARITHMETIC. 

12.  9  are  how  many  times  2  ?        How  many  2's  ? 

13.  10  are  how  many  times  2  ?        How  many  2's  ? 

14.  If  an  apple  is  worth  3  cents,  what  is  1 -third  of  it 
worth  ?     What  are  2-thirds  of  it  worth  ? 

15.  What  is  1-third  of  3?     What  are  2-thirds  of  3? 

16.  If  an  orange  is  worth  3  cents,  what  part  of  it  will 
1  cent  buy  ?     What  part  will  2  cents  buy  ? 

17.  One  is  what  part  of  3?     Ans.  One  is  I -third  of  3. 

18.  Two  is  what  part  of  3  ?     Ans.  Two  is  2-thirds  of  3, 
or  2  times  I -third  of  3. 

19.  If  a  yard  of  cloth  cost  S3,  how  much  can  you  buy 
for  84  ?     How  much  for  $5  ? 

20.  Four  are  how  many  times  3?     Ans.   Once  3  and 
1-third  of  3. 

21.  Five  are  how  many  times  3  ?      Ans.   Once  3  and 
2-thirds  of  3. 

22.  6  are  how  many  times  3  ?       7  are  how  many  3's  ? 

23.  8  are  how  many  times  3  ?       9  are  how  many  3's  ? 

24.  10  are  how  many  times  3  ?     11  are  how  many  3's  ? 

25.  If  a  lemon  is  worth  4  cents,  what  is  1 -fourth  of  it 
worth  ? 

26.  What  is  1-fourth  of  4  ?     What  are  2-fourths  of  4? 
What  are  3-fourths  of  4  ? 

27.  If  you  buy  a  yard  of  cloth  for  $4,  what  part  of  it 
can  you  buy  for  91  ?     For  82  ?     For  83  ? 

28.  What  part  of  4  is  1  ?  Ans.  1-fourth  of  4. 

29.  What  part  of  4  is  2  ?  Ans.  2-fourths  of  4. 

30.  -What  part  of  4  is  3  ?  Am.  ^-fourths  of  4. 

31.  If  a  yard  of  tape  cost  4  cents,  how  much  can  yoa 
buy  for  5  cents  ?     For  6  cents  ?     For  7  cents  ? 

32.  Five  are   how  many  times  4?      Ans.   Once  4  and 
one-fourth  of  4. 

33.  Six  are    how  many  times  4  ?      Ans.   Once  4  and 
two-fourths  of  4. 


PARTS    OF    NUMBERS.  53 

34.  Seven  are  how  many  times  4?     Ans.   Once  4  and 
three-fourths  of  4. 

35.  8  are  how  many  times  4  ?        9  are  how  many  4's  ? 

36.  10  are  how  many  times  4  ?  11  are  how  many  4's  ? 

37.  12  are  how  many  times  4  ?  13  are  how  many  4's  ? 

38.  14  are  how  many  times  4  ?  15  are  how  many  4's  ? 

39.  16  are  how  many  times  4  ?  17  are  how  many  4's  f 

40.  18  are  how  many  times  4  ?  19  are  how  many  4's  ? 

41.  20  are  how  many  times  4  ?  21  are  how  many  4's  ? 

42.  If  a  melon  is  worth  5  cents,  what  is  one-fifth  of 
it  worth?     What  are  two-fifths  worth?     What  are  three- 
fifths  worth  ?     What  are  four-fifths  worth  ? 

43.  What  is  1-fifth  of  5  ?    2-fifths  of  5  ?    3-fifths  of 
57    4-fifthsof5? 

44.  If  a  melon  is  worth  5  cents,  what  part  of  it  can 
you  buy  for  1  cent  ?     For  2  cents  ?     For  3  ?     For  4  ? 

45.  1  is  what  part  of  5  ?  Ans.  I  -fifth    of  5. 

46.  2  is  what  part  of  5  ?  Ans.  2-fifths  of  5. 

47.  3  is  what  part  of  5  ?  Ans.  ^-fifths  of  5. 

48.  4  is  what  part  of  5  ?  Ans.  ^-fifths  of  5. 

49.  If  flour  is  worth  $5  a  barrel,  how  many  barrels 
can  you  buy  for  $6  ?  For  $7  ?  For  $8  ?  For  $9  ?  For 
810?  For 


50.  6  are  how  many  times  5  ?  Once  5  and  1-fifth  of  5. 

51.  7  are  how  many  tinies-5  ?  Once  5  and  2-fifthsof5. 

52.  8  are  how  many  times  5  ?  Once  5  and  3-fifths  of  5. 

53.  9  are  how  many  times  5  ?  10  are  how  many  times  5? 

54.  11  are  how  many  times  5?  12  are  how  many  times  5? 

55.  If  1  barrel  of  apples  cost  $6,  what  is  1  -sixth  worth? 
What  2-sixths  ?     What  3-sixths  ?    4-sixths  ?    5-sixths  ? 

56.  What  is  1-sixth  of  6?     What  are  2-sixths  of  6? 
S-sixths  of  6?    4-sixths  of  6?    5-sixths  of  6? 

2d  Bk.          4 


54  KAY'S    INTELLECTUAL    ARITHMETIC. 

57.  One  is  what  part  of  6  ?     2  is  what  part  of  6  ?     3  is? 
what  part  of  6  ?     4  is  what  part  of  6  ?     5  ? 

58.  How  many  yards  of  cloth  at  $6  a  yard,  can  you 
buy  for  §7?     For  §8?     For  $9?     For  $10?     For  811  ? 
For  $12  ?     For  813  ?     For  $14  ? 

59.  Seven  are  how  many  times  6  ?     Ans.   One  time  6 
&nd  I -sixth  of  6. 

60.  8  are  how  many  times  6  ?       8  are  how  many  6's  ? 

61.  9  are  how  many  times  6  ?     10  are  how  many  6's? 

62.  11  are  how  many  times  6?     12  are  how  many  6's? 

63.  13  are  how  many  times  6  ?     14  are  how  many  6Ts  ? 

64.  15  are  how  many  times  6  ?     16  are  how  many  6's? 

65.  At  7  cents  a  yard  for  ribbon,   what   is    1 -seventh 
worth  ?     What  are  2-sevenths  worth  ?     What  are  3-sev- 
enths?     4-sevenths?     5-sevenths?     6-sevenths? 

66.  What  is  1 -seventh  of  7  ?     What  are  2-sevenths  of 
7  ?      What  are   3-sevenths  ?      4-sevenths  ?      5-sevenths  ? 
6-sevenths  ? 

67.  One  is  what  part  of  7  ?     2  is  what  part  of  7?     3 
is  what  part  of  7  ?     4  is  what  part  of  7  ?     5  ?     6  ? 

68.  How  many  barrels  of  flour  at  $7  a  barrel,  can  you 
buy  for  $8?     For  $9?     For  $10?     For  $11?     For  $12? 
For  $13?     For  $14?     For  $15  ?     For  $16? 

69.  8  are  how  many  times  7  ?       9  are  how  many  7's  ? 

70.  10  are  how  many  times  7  ?  11  are  how  many  7's  ? 

71.  12  are  how  many  times  7  ?  13  are  how  many  7's  ? 

72.  15  are  how  many  times  7  ?  16  are  how  many  7's  ? 

73.  17  are  how  many  times  7  ?  18  are  how  many  7's  ? 

74.  If  a  melon  cost  8  cents,  what   is  1 -eighth  worth? 
What  are  2-eighths  worth  ?     3-eighths  worth  ?     4-eighths 
worth?     5-eighths?     6-eighths?     7-eighths? 

75.  What  is  1 -eighth  of  8  ?     What  are  2-eighths  of  8T 
3-eighths   of   8?      4-eighths?      5-eightlw?      6-eighths? 
7-eighths  ?     8-eighths  ?     9-eighths  ?     10-eigfcths  ? 


PARTS    OF    NUMBERS.  55 

76.  One  is  what  part  of  8  ?     2  is  what  part  of  8  ?     3  is 
what  part  of  8  ?     4  is  what  part  of  8  ?     5  is  what  part 
of  8  ?     6  is  what  part  of  8  ?     7  ? 

77.  How  many  gallons  of  beer,  at  8  cents  a  gallon,  can 
you  buy  for  9  cents  ?      For  10  cents  ?      For  11  cents  ? 
For  12  cents  ?     For  13  cents  ?     For  14  cents  ?     Foj  15 
cents  ?     16  cents  ?     17  cents  ?     18  cents  ? 

78.  9  are  how  many  times  8  ?  Once  8  and  \-eiglitTi  of  8. 

79.  10  are  how  man^  times  8  ?  11  are  how  many  8's  ? 

80.  12  are  how  many  times  8  ?  13  are  how  many  8's  ? 

81.  14  are  how  many  times  8  ?  15  are  how  many  8's  ? 

82.  16  are  how  many  times  8  ?  17  are  how  many  8's  ? 

83.  Eighteen  are  how  many  times  8  ?     Ans.   Two  time* 
8  and  2-ciglitlis  of  8. 

84.  If  an  orange   cost   9  cents,  what  is  1 -ninth  of  it 
worth  ?     What  are  2-ninths  worth  ?    3-ninths  ?    4-ninths  ? 
5-ninths  ?     6-ninths  ?     7-ninths  ?     8-ninths  ? 

85.  What  is  1-ninth  of  9  ?     What  are  2-ninths  of  9  ? 
What  are  3-ninths  of  9  ?     4-ninths  of  9  ?     5-ninths  of  9? 
6-ninths  of  9  ?     7-ninths  of  9  ?     8-ninths  of  9  ? 

86.  One  is  what  part  of  9  ?     2  is  what  part  of  9  ?     3  is 
•what  part  of  9  ?     4  is  what  part  of  9  ?     5  is  what  part 
of  9  ?     6  ?     7  ?     8  ? 

87.  If  cloth  is  $9  a  yard,  how  much  can  you  buy  for 
$10?     For  $11?     For  $12?     For  $13  ?     For  $14?     For 
$15?      For   $16?      For   $17?      For   $18?      For   $19? 
For  $20  ? 

88.  If  apples  cost  10  cents  a  bushel,  what  is  1 -tenth 
of   a   bushel  worth?      2-tenths?      3 -tenths  ?      4-tenths  ? 

-  5-tenths  ?     6-tenths  ?     7-tenths  ?     8-tenths  ?     9-tenths  ? 

89.  One  is  what  part  of  10?      2  is  what  part  of  10? 
3  is  what  part  of  10  ?    4  is  what  part  of  10  ?     5  is  what 
part  of  10  ?     6  is  what  part  of  10  ?     7  is  what  part  of 
10  ?     8  is  what  part  of  10  ?     9  ?     11  ? 


56  RAY'S   INTELLECTUAL   ARITHMETIC. 

90.  When  cloth  is  §10  a  yard,  how  much  can  you  buy 
for  §11  ?      For  §12  ?      For  §13  ?     For  §14  ?     For  §15  ? 
For  §16  ?     For  §17  ?      For  §18  ?     For  §19  ? 

91.  What  do  you  understand  by  1 -fifth  of  any  thing? 
By  3-fifths?     By  4-fifths? 

92.  What   do   you   understand  by   3-sevenths    of  any 
thing?     By  2-eighths? 

93.  If  I  cut  an  apple  into  6  equal  parts,  and  give  you 
4,  what  part  of  the  whole  apple  would  I  give  you  ? 

94.  If  I  divide  an  orange  into  8  equal  parts,  and  give 
you  5,  what  part  of  the  orange  would  I  give  you  ? 

95.  What  is  meant  by  1 -ninth  of  any  thing  ?     2 -ninths  ? 

96.  What  is  meant  by  1-tenth  of  any  thing  ?     2-tenths  ? 

LESSON  III. 

1.  If  you  had  7  cents,    how  many   cakes  could  you 
buy  at  2  cents  each?     Ans.  3  cakes  and  1-half. 

2.  Seven  are  how  many  times  2  ?     Ans.  3  times  2  and 
l-half  of  2. 

3.  If  you  had  11  cents,  how  many  pears  could  you 
buy  at  2  cents  each  ?     At  3  cents  ? 

4.  Eleven  are  how  many  times  2  ?     How  many  times 
3  ?     Ans.  Three  times  3  and  2-thirds  of  3. 

5.  If  you  had  15  cents,  how  many  cakes  could  you 
buy  at  2  cents  each  ?     At  3  cents  ?     At  4  cents  ?     At  5  ? 

6.  Fifteen  are  how  many  times  2?     3?    4?     5? 

7.  If  you  had  17  cents,  how  many  oranges  could  you 
buy  at  2  cents  each  ?     At  3  cents  each  ?     At  4  cents  each  ? 
At  5  cents  ?     At  6  cents  ?     At  7  cents  ? 

8.  24  are  how  many  times  2  ?  3?  4?  5?  6?  7?  8? 

9.  25  are  how  many  times  2  ?  3?  4?  5?  6?  7?  8? 
10.    26  are  how  many  times  3  ?  4?  5?  6?  7?  8?  9? 
11. .  27  are  how  many  times  3  ?  4?  5?  6?  7?  8?  9? 
12.   28  are  how  many  times  3  ?  4  5?  6?  7?  8?  9? 


PARTS   OF    NUMBERS.  5-7 

13.  29  are  how  many  times  3  ?  4?  5?  6?  7?  8?  9* 

14.  30  are  how  many  times  3?  ,4?  5?  6?  7?  8?  9? 

15.  31  are  how  many  times  3?  4?  5  ?  6?  7?  8?  9? 

16.  32  are  how  many  times  3?  4?  5?  6?  7?  8?  9? 

17.  33  are  how  many  times  3  ?  4?  5?  6?  7?  8?  9? 

18.  34  are  how  many  times  3?  4?  5?  6?  7?  8?  9? 

19.  35  are  how  many  times  3  ?  4?  5?  6?  7?  8?  9? 

20.  36  are  how  many  times  3?  4?  5?  6?  7?  8?  9? 

21.  37  are  how  many  times  3  ?  4?  5?  6?  7?  8?  9? 

22.  38  are  how  many  times  3?  4?  5?  6?  7?  8?  9? 

23.  39  are  how  many  times  4?  5?  6?  7?  8?  9? 

24.  40  are  how  many  times  4?  5?  6?  7?  8?  9? 

25.  If  you  had  41  cents,  how  many  oranges  could  you 
buy  at  4  cents  each  ?      At  5  cents  eac-h  ?      At  6  cents  ? 
At  7  cents  ?     At  8  cents  ?     At  9  cents  ?     At  10  cents  ? 

26.  42  are  how  many  times  4?     5?     6?    7?     8?     9? 

27.  43  are  how  many  times  4  ?     5?     6?     7?     8?     9? 

28.  44  are  how  many  times  4  ?  5?  6?  7?  8?  9? 

29.  45  are  how  many  times  4?  5?  6?  7?  8?  9? 

30.  46  are  how  many  times  4?  5?  6?  7?  8?  9? 

31.  47  are  how  many  times  4?  5?  6?  7?  8?  9? 

32.  48  are  how  many  times  4?  5?  6?  7?  8?  9? 

33.  49  are  how  many  times  4?  5?  6?  7?  8?  9? 

34.  If  you  had  56  cents,  how  many  peaches  could  you 
buy  at  5  cents  each  ?      At  6  cents  each  ?      At  7  cents  ? 
At  8  cents  ?     At  9  cents  ?     At  10  cents  ? 

35.  56  are  how  many  times  5?  6?  7?  8?  9?  10? 

36.  57  are  how  many  times  5?  6?  7?  8?  9?  10? 

37.  58  are  how  many  times  5?  6?  7?  8?  9?  10? 

38.  59  are  how  many  times  5?  6?  7?  8?  9?  10? 

39.  60  are  how  many  times  5?  6?  7?  8?  9?  10? 


58  KAY'S  INTELLECTUAL  ARITHMETIC. 

40.     61  are  how  many  times  6?     7  ?     8?     9?     10? 


*?     9?     10? 


41.  63  are  how  many  times  6  ?  7  ? 

42.  65  are  how  many  times  6  ?  7?  8?     9?     10? 

43.  66  are  how  many  times  6?  7?  8?     9?     10? 

44.  68  are  how  many  times  6?  7?  8?     9?     10? 

45.  69  are  how  many  times  6?  7?  8?     9?     10? 

46.  If  70    hours  be   required    to  perform   a  piece  of 
work,  in  how  many  days  can  it  be  done  by  working  6 
hours  a  day  ?     By  working  7  hours  a  day  ?     8  hours  a 
day  ?  9  hours  ?     10  hours  ? 

47.  71  are  how  many  times  6?  7?  8?       9?     10? 

48.  72  are  how  many  times  6?  7?  8?       9?     10? 

49.  74  are  how  many  times  6?'  7?  8?       9?     10? 

50.  76  are  how  many  times  6?  7?  8?       9?     10? 

51.  77  are  how.  many  times  6  ?  7?  8?       9?     10? 

52.  79  are  how  many  times  7?  8?  9?     10?     11? 

53.  80  are  how  many  times  7  ?  8?  9?     10?     11? 

54.  81  are  how  many  times  7  ?  8  ?  9  ?     10  ?     11  ? 

55.  82  are  how  many  times  7  ?  8?  9?     10?     11? 

56.  83  are  how  many  times  7  ?  8  ?  9  ?     10  ?     11  ? 

57.  85  are  how  many  times  7  ?  8?  9?^    10?     11? 

58 .  86  are  how  many  times  7?  8?  9?     10?     11? 

59.  87  are  how  many  times  7?  8?  9?     10?     11? 

60.  88  are  how  many  times  7  ?  8  ?  9  ?     10  ?     11  ? 

61.  89  are  how  many  times  7  ?  8?  9?     10?     11? 

62.  90  are  how  many  times  7?  8?  9?     10?     11? 

63.  91  are  how  many  times  8  ?  9  ?  10  ?     11  ?     12  ? 

64.  92  are  how  many  times  8  ?  9  ?  10  ?     11  ?     12  ? 

65.  93  are  how  many  times  8  ?  9?  10?     11?     12? 

66.  94  are  how  many  times  8  ?  9?  10?     11?     12? 

67.  95  are  how  many  times  8  ?  9  ?  10  ?     11  ?     12  ? 

68.  96  are  how  many  times  8  ?  9  ?  10  ?     11  ?     12  ? 


PARTS   OP    NUMBERS.  59 

69.  98  are  how  many  times  8  ?     9  ?     10  ?  11  ?  12  ? 

70.  99  are  how  many  times  8  ?     9  ?     10  ?  11  ?  121 

71.  100  are  how  many  times  8  ?     9  ?     10  ?  11  ?  12  ? 


SECTION  VIIL 

LESSON  I. 

1.  At  2  cents  each,  what  will  2  apples  and  1-half  of 
an  apple  cost? 

ANALYSIS. — If  1  apple  cost  2  cents,  2  apples  will  cost 
twice  2  cents,  which  are  4  cents;  and  1-half  of  an  apple 
will  cost  1-half  of  2  cents,  which  is  1  cent;  and  4  cents  and 
1  cent  are  5  cents ;  therefore,  if  1  apple  cost  2  cents,  2  apples 
and  l-half  will  cost  5  cents. 

2.  Two  times  2  and  1-half  of  2  are  how  many? 

3.  At  2  cents  each,  what  will  3  pears  and  1-half  of  a 
lear  cost? 

4.  Three  times  2  and  1-half  of  2  are  how  many? 

5.  At  $3  a  yard,  what  will  3  yards  and  1-third  of  a 
yard  of  cloth  cost? 

6.  Three  times  3  and  1-third  of  3  are  how  many? 

7.  At  $3  a  barrel,  what  will  3  barrels  and  2-thirds  of 
a  barrel  of  flour  cost? 

SUGGESTION.— To  find  2-thirds  of  any  number,  first  find  1-third, 
then  multiply  by  2. 

8.  Three  times  3  and  2-thirds  of  3  are  how  many? 

9.  At  S3  a  yard,  what  will  4  yards  and  2-thirds  of  a 
yard  of  cloth  cost? 

ANALYSIS. — If  1  yard  cost  $3,  4  yards  will  cost  4  times 
$3,  which  are  $12;  and  1-third  of  a  yard  will  cost  1-third 
of  $3,  winch  is  $1 ;  and  2-thirds  of  a  yard  will  cost  twice 
as  much  as  1-third,  that  is,  twice  $1,  which  are  $2;  and  $12 
and  $2  are  $14.  Ans.  $14. 


bU  KAY'S    INTELLECTUAL   ARITHMETIC. 

10.  If  an  orange   cost  4  cents,   what  will   3  oranges 
and  1 -fourth  of  an  orange  cost  ? 

11.  3  times  4  and  2 -fourths  of  4  are  how  many? 

12.  4  times  4  and  3-fourths  of  4  are  how  many  ? 

,    13.  At  85  a  barrel,  what  will  4  barrels  and  2-fifths  of 
a  barrel  of  flour  cost? 

14.  4  times  5  and  2-fifths  of  5  are  how  many  ? 

15.  5  times  5  and  3-fifths  of  5  are  how  many  ? 

16.  If  a  man  spend  $6  a  week,  how  much  will  he  spend 
in  4  weeks  and  1  -sixth  of  a  week  ? 

17.  4  times  6  and  2-sixths  of  6  are  how  many  ? 

18.  5  times  6  and  3-sixths  of  6  are  how  many  ? 

19.  6  times  6  and  4-sixths  of  6  are  how  many  ? 

20.  7  times  6  and  5-sixths  of  6  are  how  many  ? 

21.  At  7  cents   a   yard,   how  much  will  3  yards  and 
2-sevenths  of  a  yard  of  tape  cost  ? 

22.  4  times  7  and  3-sevenths  of  7  are  how  many  ? 

23.  5  times  7  and  4-sevenths  of  7  are  how  many  ? 

24.  6  times  7  and  6-sevenths  of  7  are  how  many  ? 

25.  If  oranges  are  worth  8  cents  each,  how  much  are 
3  oranges  and  2-eighths  of  an  orange  worth  ? 

26.  4  times  8  and  3-eighths  of  8  are  how  many  ? 

27.  5  times  8  and  4-eighths  of  8  are  how  many  ? 

28.  6  times  8  and  7-eighths  of  8  are  how  many  ? 

29.  If  a  yard  of  muslin  cost  9  cents,  what  will  2  yards 
and  2-ninths  of  a  yard  cost  ? 

30.  3  times  9  and  5 -ninths  of  9  are  how  many  ? 

31.  5  times  9  and  6 -ninths  of  9  are  how  many  ? 

32.  7  times  9  and  8-ninths  of  9  are  how  many  ? 

33.  If  a  pound  of  sugar   cost   10   cents,  what  will  2 
pounds  and  2-tenths  of  a  pound  cost  ? 


PARTS   OF    NUMBERS.  Q\ 

34.  2  times  10  and  3-tenths        of  10  are  how  many  ? 

35.  5  times  10  and  6-tenths        of  10  are  how  many  ? 

36.  6  times  10  and  8-tenths        of  10  are  how  many  ? 

37.  9  times  10  and  7-tenths        of  10  are  how  many  ? 

38.  10  times    9  and  8-ninths        of    9  are  how  many  ? 

39.  12  times  11  and  9 -elevenths  of  11  are  how  many  ? 


SECTION  IX. 

LESSON  I. 

1.  A  boy  bought  3  apples  at  4  cents  each :  how  much 
did  they  cost?     He  paid  for  them  with  oranges,  at  6  cents 
each  :  how  many  oranges  did  it  take  ? 

ANALYSIS. — First.  3  apples  cost  3  times  the  price  of  1  apple, 
that  is,  3  times  4  cents,  or  \2  cents.  Second.  It  took  as  many 
oranges  as  6  cents  are  contained  times  in  12  cents;  6  cents 
are  contained  in  12  cents  2  times,  Ans.  2  oranges. 

2.  A  man  bought  3  yards  of  cloth  at  $4  a  yard :  how 
many  dollars  did  it  cost?     He  paid  for  it  with  cider,  at 
$2  a  barrel :  how  many  barrels  did  it  take  ? 

3.  A  boy  bought  8  apples  at  2  cents  each :  how  much 
did  they  cost?     He  paid  for  them  with  pears,  at  4  cents 
each  :  how  many  did  it  take  ? 

4.  Bought  9  marbles  at  2  cents  each ;  paid  for  them 
with  tops,  at  6  cents  each :  how  many  did  it  take  ? 

5.  Bought  10  yards  of  cloth  at  $2  a  yard;  paid  for 
it  with  flour,  at  $4  a  barrel :  how  many  bar,  did  it  take? 

6.  Bought  8  pints  of  cherries  at  3  cents  a  pint  j  paid 
for  them  with  apples,  at  6  cents  a  dozen :  how  many  dozen 
did  it  take  ? 

7.  How  many  barrels  of  flour,  at  $3  a  barrel,  must  be 
given  for  2  yards  of  cloth,  at  $7  a  yard  ? 


S2  RAY'S    INTELLECTUAL    ARITHMETIC. 

8.  5  times    3  are  how  many  times  4  ?     6?       7  ?       8? 

9.  4  times     4  are  how  many  times  3?     5?       6?       7  ? 

10.  5  times    4  are  how  many  times  3?  6?  7?  8? 

11.  2  times  11  are  how  many  times  3?  4?  5?  6? 

12.  5  times     5  are  how  many  times  3?  4?  6?  7? 

13.  8  times    4  are  how  many  times  3  ?  5?  6?  7? 

14.  7  times     5  are  how  many  times  4?  6?  8?  9? 

15.  6  times    7  are  how  many  times  4?  5?  8?  9? 

16.  6  times     9  are  how  many  times  5?  7?  8?  10? 

17.  8  times    7  are  how  many  times  5?  6?  9?  10? 

18.  6  times  10  are  how  many  times  5  ?  7  ?  8  ?  10  ? 

19.  9  times    7  are  how  many  times  6  ?  8  ?  10  ?  11  ? 

20.  8  times    8  are  how  many  times  7  ?  9  ?  10  ?  11  ? 

LESSON  II. 

1.  Bought  4  boxes  and  3-fifths  of  a  box  of  raisins,  at 
85  a  box  :  how  much  did  they  cost  ?     Paid  for  them  with 
flour,  at  §6  a  barrel :  how  many  did  it  take  ? 

ANALYSIS. — It  will  take  as  many  barrels  of  flour  as  $6,  the 
price  of  1  barrel,  are  contained  times  in  the  cost  of  4  boxes 
and  %-fifths  of  a  box  of  raisins,  at  $5  a  box. 

2.  Bought  4  gallons  and  4-sixths  of  a  gallon  of  wine, 
for  §6  a  gallon,  and  paid  for  it  with  raisins,  at  §5  a  box : 
how  many  boxes  did  it  take  ? 

3.  Bought  5  kegs  and  4-sevenths  of  a  keg  of  tobacco, 
for  §7  a  keg,   and  paid  for   it  with  paper,  at  86  a  ream : 
how  many  reams  did  it  take  ? 

4.  Five   times    5    and    3-fifths    of   5    are    how   many 
times  3?     4?     6?     7?     8?     9?     10? 

5.  Seven  times  4  and   3-fourths  of  4  are  how  many 
times  3?     5?     6?     8?     9?     10? 

6.  Five  times   6   and    5-sixths    of   6    are   how   many 
times  3?    4?     7?     8?     9?     10? 


PARTS    SF    NUMBERS.  63 

7    Seven   times   5    and   3-fifths  of  5   are  how  many 
times  4?     6?     8?     9?     10? 

8.  Five  times  8  and   1 -eighth   of   8    are    how   many 
times  4?     6?     7?     9?     10? 

9.  Seven   times   6   and  2 -sixths  of   6   are  how  many 
times  5?     8?     9?     10? 

10.  Nine   times   5    and    2 -fifths    of   5    are  how    many 
times  6?     7?     8?     10? 

11.  Nine   times    5    and   4-fifths    of   5    are   how 
times  6?     7?     9?     10? 

12.  Six  times  8  and  3 -eighths    of   8    are    how    many 
times  5?     7?     9?     10? 

13.  Seven  times  7  and  4-sevenths  of  7  are  how  many 
times  5?     6?     8?     9?     10? 

14.  Nine  times  6  and   3-sixths  of   6   are   how   many 
times  5?     7?     8?     10? 

15.  Eight  times  7  and  3 -sevenths  of  7  are  how  many 
times  5?     6?     9?     10? 


LESSON  III. 

1.  Bought  4  apples  at  3  cents  each:  paid  for  them 
lemons  at  6  cents  each :  how  many  did  it  take  ? 

2.  Bought  7  yards  of  tape,  at  2  cents  a  yard  :    how 
many  pears  at  3  cents  each,  will  it  take  to  pay  for  it  ? 

3.  If  2  apples  cost  4  cents,  what  cost  3  apples  ? 

ANALYSIS. — Three- apples  will  cost  3  times  as  much  as  1 
apple,  and  1  apple  will  cost  l-half  as  much  as  2  apples,  that 
is,  l-half  of  4  cents,  \-half  of  4  cents  is  2  cents,  and  3  times 
2  cents  are  6  cents.  Ans.  6  cents. 

4.  If  3  yards  of  cloth  cost  $9,  what  cost  4  yards  ? 

5.  If  3  oranges  cost  15  cents,  what  cost  5  oranges? 

6.  If  4  barrels  of  flour  cost  $24,  what  cost  7  barrels  ? 

7.  If  2  kegs  of  lard  cost  $8,  what  cost  9  kegs  ?  11  kegs  ? 


G4  RAY'S   INTELLECTUAL   ARITHMETIC. 

8.  If  5  dozen  eggs  cost  35  cents,  what  will  3  dozen 
eost  ?     What  will  8  dozen  cost  ? 

9.  If  a  man  get  §14  for  7  days'  work,  how  much  will 
he  get  for  9  days  ?     For  3  days  ? 

10.  Bought  4  yards  and  2-thirds  of  a  yard  of  cloth,  at 
$3  a  yard,  and  paid  for  it  with  cheese,  at  87  a  hundred 
weight :  how  many  hundred  weight  did  it  take  ? 

11.  Bought  4  pounds  and  4-fifths  of  a  pound  of  nails, 
at  5   cents  a  pound,  and  paid  for  them  with   eggs,   at  3 
cents  a  dozen  :  how  many  dozen  did  it  take  ? 

12.  Bought  7   pounds   and   5-sevenths  of  a  pound  of 
sugar,  at  7  cents  a  p<?>md,  and  paid  for  it  with  chickens^ 
at  9  cents  each  :  how      any  did  it  take  ? 

13.  Bought  9  pounds   and   2-sevenths  of  a  pound  of 
sugar,  at  7  cents  a  poand,  and  paid  for  it  with  eggs,  at  6 
eents  a  dozen :  how  many  dozen  did  it  take  ? 

14.  How  many  pounds,  at  7  cents  a  pound,  must  you 
give  for  8  and  2-ninth  yards,  at  9  cents  a  yard  ? 

15.  How  many  barrels,  at  $6  a  barrel,  must  be  given  in 
exchange  for  4  and  5 -seventh  yards,  at  §7  a  yard  ? 

16.  Bought  5  pounds   and  3-sevenths  of  a  pound  of 
butter,  at  7  cents  a  pound,  and  paid  for  it  with  raisins,  at 
6  cents  a  pound :  how  many  pounds  did  it  take  ? 

17.  How  many  apples,   at   2  cents   each,  can  you  buy 
for  6  cents?     How  many  for  15  cents? 

18.  How  many  pears,  at  3    for  7  cents,  can  you  buy 
for  21  cents  ?     For  35  cents  ? 

19.  If  6  pears  are  worth  2  oranges,  how  many  oranges 
ean  you  buy  for  21  pears  ? 

20.  A  man  bought  15  yards  of  cloth,  at  the  rate  of  3 
yards  for  $5  :   how  many  dollars  did  it  cost  ? 

21.  If  a  man  receive  §5  for  4  days'  work,  how  many 
dollars  will  he  get  for  12  days'  work  ? 

22.  How   many  pears,  at  3  for  5  cents,  can  you  buy 
for  25  cents  ? 


PARTS   OF   NUMBERS.  65 

23.  If  2  pears  are  worth  6  cents,  how  many  pears  must 
be  given  for  4  oranges,  at  6  cents  each  ? 

24°.  What  will  be  the  cost  of  3  barrels  and  9-elevenths 
of  a  barrel  of  sugar,  at  $11  a  barrel? 

25.  What  will  be  the  cost  of  3  boxes  and  4-fifths  of 
a  box  of  butter,  at  $5  a  box  ? 

26.  Find  the  cost  of  4  and  5-sixth  tons  of  hay,  at  $6 
a  ton. 

27.  How   many   dozen   eggs,  at  12  cents  a  dozen,  will 
pay  for  10  and  10-eleventh  pounds  of  sugar,  at  11  cents 
a  pound  ? 


SECTION  X. 

LESSON  I. 

1.  If  1-half  of  an  orange  cost  3  cents,  what  will  the 
whole  orange  cost? 

ANALYSIS.  —  1  orange  will  cost  twice  as  much  as  \-Jialf  of 
an  orange,  that  is,  twice  8  cents.  But  twice  3  cents  are  6  cents  ; 
therefore,  if  1-half  of  an  orange  cost  3  cents,  the  whole  orange 
l  cost  6  cents. 


2.  Three  is  1-half  of  what  number? 

ANALYSIS.—  3  is   I-half  of  twice  3;  but  twice  3  are  6; 
Cher  ef  ore,  3  is  l-half  of  6. 

3.  If  1  -fourth  of  a  barrel  of  cider  cost  $2,  what  will 
a  barrel  cost? 

4.  Two  is  1  -fourth  of  what  number? 

5.  If  a  man  can  walk  2  miles  in  1  -third  of  an  hour, 
how  far  can  he  walk  in  an  hour  ? 

6.  Two  is  1  -third  of  what  number? 

7.  The  age   of  Charles  is   1  -third  that  of  Thomas; 
Charles  is  4  years  old  :  how  old  is  Thomas  ? 


6o    •  AAY'S   INTELLECTUAL  ARITHMETIC. 

8.  Four  cents  is  the  1-third  of  what  number  of  cents! 

9.  David  has   5   marbles,   which  is  only   1 -fourth  as 
many  as  Henry  :  how  many  has  Henry  ? 

ANALYSIS. — 5  is   I -fourth  of  four  times  5;   but  4  times  5 
are  20:  therefore,  5  is  1-fourth  of  20. 

10.  Six  is  1 -fourth  of  what  number  ?     8  is  1 -fourth  of 
?rhat  number  ?     9  is  1 -fourth  of  what? 

11.  Six  is  1-third  of  what  number?     7  is  1-third  of 
what  number?     9  is  1-third  of  what? 

12.  Three  is  1-fifth  of  what  number?     5  is  1-fifth  of 
what  number?     9  is  1-fifth  of  what? 

13.  Five  is  1-sixth  of  what  number?     7  is  1-sixth  of 
what  number?.   9  is  1-sixth  of  what? 

14.  Two  is  1 -seventh  of  what  number?     5  is  1 -seventh 
of  what  number  ?     8  is  1 -seventh  of  what? 

15.  Five  is  1-eighth  of  what  number?     7  is  1-eighth 
of  what  number?     9  is  1-eighth  of  what? 

16.  Seven  is  1-ninth  of  what  number?     8  is  1-ninth 
of  what  number?     9  is  1-ninth  of  what? 


LESSON   II. 

1.  James  had  4  apples,  and  gave  his  brother  1-half 
of  them  :  how  many  did  he  give  him  ? 

2.  If  you  divide   6   apples   equally  between   2  boys, 
what  part  of  them  must  each  have?     Ans.  \-lialf. 

3.  If  you   divide   6   apples  equally  between  3  boys, 
what  part  of  them  must  each  have?     Ans.  1-third. 

4.  If  3  yards  of  cloth  cost  $9,  what  part  of  §9  will  1 
yard  cost  ?     What  part  of  89  will  2  yards  cost  ? 

5.  If  4  oranges  cost  12  cents,  what  part  of  12  cents 
will  1  orange  cost?     What  part  of  12  cents  will  2  cost? 
What  part  will  3  cost? 

6.  What   is    1 -fourth    of   12?     What   are    2-fourths 
c<f  12?    3-fourthsof  12? 


PAETS    OF    LUMBERS.  67 

7.  If  5  barrels  of  flour  cost  $30,  what  part  of  $30 
will  1  barrel  cost?     What  part  of  $30  will  2  barrels  cost? 
What  part  will  3  bar.  cost  ?     What  part  will  4  cost  ? 

8.  WThat  is  1-fifth  of  30?     What  are  2-fifths  of  30? 
3-fifths  of  30?    4-fifths  of  30  ? 

9.  If  1   apple   cost   4   cents,  what  will   1-half  of  an 
apple  cost  ?     What  will  3-halves  cost  ? 

10.  If  2  oranges  cost  8  cents,  what  part  of  8  cents 
will  1  orange  cost  ?     3  oranges  ?     5  ? 

11.  If  3  barrels  of  cider  cost  $12,  what  part  of  $12  will 
1  barrel  cost?     What  part  of  $12  will  2  barrels  cost?     4 
barrels  ?     5  barrels  ? 

12.  What  is  1 -third  of  12  ?     What  are  2-thirds  of  12? 
4-thirds  of  12?     5-thirds  of  12? 

13.  What  is  1 -fourth  of  24?     2-fourths?     3-fourths? 
5-fourths  ?     6-fourths  ?     7-fourths ?     9-fourths  ? 

14.  What    is    1-fifth    of    25?       2-fifths?      3-fifths? 
4-fifths?     6-fifths?    7-fifths?    8-fifths?     9  -  fifths  ? 

15.  What    is    1-sixth    of    24?      2-sixths  ?      3-sixths  ? 
4-sixths  ?     5-sixths  ?     6-sixths  ?     7-sixths  ? 

16.  What  is    1-seventh    of   56?     2-sevenths?     3-sev- 
enths?  4-sevenths?  5 -sevenths?   7-sevenths? 

17.  What  is  1 -eighth  of  72  ?     2-eighths  ?     5-eighths? 
8-eighths  ?     10-eighths  ?     11-eighths  ? 

18.  What   is   1-ninth    of   54?      3-ninths  ?     5-ninths? 
7-ninths  ?     9-ninths  ?    11-ninths  ?     12-ninths  ? 

19.  What  is   1-tenth   of   60?      3-tenths  ?      7-tenths? 
9-tenths?     11-tenths? 

20.  What  are  2-thirds  of  6  ? 

ANALYSIS. —  Two-thirds  of  6  are  twice  l-third.     One-third 
vf  6  is  2,  and   twice  2  are  4 ;  therefore,  2-thirds  of  6  are  4. 

21.  What  are  2-thirds  of  12?         3-fourths  of  12? 

22.  What  are  4-fifths    of  20  ?         5-sixths     of  30  ? 

23.  What  is     1-ninth  of  27  ?         1 -fourth    of  36  ? 


RAY'S    INTELLECTUAL   ARITHMETIC. 


24. 

What 

are 

3-sevenths 

of 

28? 

2-fifths 

of 

20? 

25. 

What 

are 

4-sixths 

of 

24? 

3-fourths 

of 

20? 

26. 

What 

are 

5-ninths 

of 

18? 

6  -sevenths 

of 

21? 

27. 
28. 
29. 

What 
What 
What 

are 
are 
are 

6  -sevenths 
5  -eighths 
6  -sevenths 

of 
of 
of 

49? 
40? 
63? 

3-eighths 
4-ninths 
7-eighths 

of 
of 
of 

24? 
54? 

56? 

30. 

What 

are 

3  -halves 

of 

18? 

4-thirds 

of 

24? 

31. 

What 

are 

7-fourths 

of 

12? 

8-fifths 

of 

30? 

32. 

What 

are 

8-sixths 

of 

42? 

9-sevenths 

of 

63? 

33. 

34. 

What 
What 

are 
are 

9-eighths 
11  -tenths 

of 
of 

56? 
50? 

10  -ninths 
12-tenths 

of 
of 

81? 
40  > 

LESSON  III. 

1.  A  man  having  12  bushels  of  grain,  divided  5-sixths 
of  it  equally  among  3  poor  persons  :  how  many  bushels 
did  each  receive? 

2.  Five-sixths  of  12  are  how  many  times  3? 

3.  A  boy  having  25  apples,  kept  1 -fifth  himself,  and 
divided  *he  other  4-fifths  equally  among  6  of  his  com- 
panions •  how  many  did  each  receive  ? 


4. 

4-fifths 

of 

25 

are 

how 

many 

times 

6? 

5. 

3-fourths 

Of 

24 

are 

how 

many 

times 

9? 

6. 

7-fourths 

of 

24 

are 

how 

many 

times 

8? 

7. 

8-thirds 

of 

18 

are 

how 

many 

times 

6? 

8. 

7-thirds 

of 

27 

are 

how 

many 

times 

10? 

9, 

3-fifths 

of 

60 

are 

how 

many 

times 

7? 

10. 

5-sixths 

of 

54 

are 

how 

many 

times 

8? 

11. 

8-sixths 

of 

48 

are 

how 

many 

times 

9? 

12. 

3-sevenths 

of 

56 

are 

how 

many 

times 

9? 

13. 

9-sevenths 

of 

63 

are 

how 

many 

times 

10? 

14. 

5  -eighths 

of 

64 

are 

how 

many 

times 

6? 

15. 

9-eighths 

of 

40 

are 

how 

many 

times 

7? 

PARTS  OF   NUMBERS.  69 

16.  11 -sevenths  of  49  are  how  many  times    8  ? 

17.  3-ninths       of   54  are  how  many  times     7  ? 

18.  10 -ninths      of   63  are  how  many  times     8  ? 

19.  8-ninths      of   54  are  how  many  times     5  ? 

20.  9-sevenths  of  42  are  how  many  times     8  ? 

LESSON   IV. 

1.  James   gave    his    brother    2    apples,   which    were 
Vthird  of  all  he  had:  how  many  had  he  left? 

2.  Thomas   gave   his    brother    5    cents,    which    were 
1 -fourth  of  all  he  had :  how  many  had  he  ? 

3.  If,  in  traveling,  I  walk  1 -fifth  of  my  journey  in  2 
hours,  at  that  rate,  in  what  time  can  I  complete  the  re- 
maining 4-fifths  ? 

4.  One  pint  is  1 -eighth  of  a  gallon  :  if  a  pint  of  wine 
cost  7  cents,  what  will  a  gallon  cost  ? 

5.  A  boy  found  a  purse  containing  $12,  and  received 
1 -sixth  of  the  money  for  returning  it  to  the  owner :  how 
much  did  he  receive? 

6.  If  $42  be   equally   divided   among  a   number   of 
men,  giving  each  man  1 -sixth  of  the  money,  how  many 
men  would  there  be,  and  what  would  each  receive  ? 

7.  Thomas  had  28  marbles:  he  gave  1 -fourth  of  them 
to  James,  and  twice  as  many  to  William  as  to  James : 
how  many  did  each  receive  ? 

8.  William  had  24  apples:  he  gave  1-half  to  Thomas, 
and  1 -third  to  James:  how  many  did  he  give  to  both? 
How  many  had  he  left? 

9.  A  boy  had  12  cents:  he  spent  1 -third  of  them  fcaf 
apples,  and  1 -fourth  for  cakes  :  how  many  had  he  left? 

10.  A  little  girl  received  20  cents  from  her  mother; 
her  brother  gave  her  1 -fifth  as  many  as  her  mother,  and 
her  sister  1-half  as  many  as  her  brother :  how  many  did 
she  then  have  ? 

2d  Bk.         '  5 


70  RAT'S   INTELLECTUAL  ARITHMETIC. 

11.  A  boy  having  40  cents  gave  3-fifths  of  them  foi 
2  arithmetics :  what  was  the  price  of  1  arithmetic  ? 

ANALYSIS. — 3-Jifths  are  3  times  \-fifih:  \-fifih  of  40  cents 
is  8  cents,  and  3  fifths  are  3  times  8  cents,  ivhich  are  24  cents. 
If  2  books  cost  24  cents,  1  book  will  cost  \-Jialf  of  24  cents> 
which  is  12  cents.  Ans.  12  cents. 

12.  James  had  14  cents,  and  gave  4-sevenths  of  them 
to  his  sister :  how  many  cents  had  he  left  ? 

13.  John   had    15    pears:    he   gave   1-third  to  Frank, 
and  3-fifths  to  Harry  :  how  many  had  he  left  ? 

14.  A  man  had  30  yards  of  cloth,  and  sold  2-fifths  of 
it  for  §48  :  how  much  was  that  a  yard? 

15.  John  had  25  cents,  and  gave  3-fifths  for  peaches, 
at  2  cents  each :  how  many  did  he  buy  ? 

16.  A  boy  having  54   chestnuts,    divided   5 -ninths  of 
them  among  3  girls :  how  many  did  each  receive  ? 

17.  A  man  had  28  barrels  of  flour,  and  sold  2-sevenths 
of  them  for  §24 :  what  was  that  a  barrel  ? 

18.  Bought  7  yards  of  cloth  for  §42 :  I  gave  3  yards 
for  9  barrels  of  cider :  how  much  was  it  a  barrel  ? 

19.  A  man  had  §40,  and  lost  3-fifths  of  them:  he  ex- 
pended the  remainder  in  flour,  at  84  a  barrel :  how  many 
barrels  did  he  buy  ? 

20.  I  had  10  oents,  and  lost  1 -fifth :  spent  the  rest  for 
apples,  at  2  cents  each  :  how  many  did  I  buy  ? 

21.  James   had   48   cents :    he   gave  3-eighths   to   his 
brother,  and    spent   the   rest   in  chestnuts,   at  5   cents  a 
quart :  how  many  quarts  of  chestnuts  did  he  buy  ? 

22.  Thomas   had    28    cents:    he    gave  1-fourth  to   his 
sister,  and  3-sevenths  to  his  brother,  and  with  the  remain 
der  he  bought  3  books  :  what  did  each  cost  ? 


•PARTS    OF   NUMBERS.  71 

SECTION  XI. 

LESSON  I. 

1.  If  2-thirds   of  a   melon   cost  4  cents,  what  will 
1 -third  cost? 

ANALYSIS. — l-third  is  I -half  of  2-thirds:   if  2-thirds  cost 
4  cents,  l-third  will  cost  l-half  of  4  cents,  or  2  cents,  Ans. 

2.  Four  is   2  times  what  number?     Ans.  Four  is  2 
times  l-half  of  4,  which  is  2. 

3.  If  3-fourths  of  a  yard  of  cloth  cost  $6,  what  will 
1 -fourth  cost? 

4.  Six  is  3  times  what  number? 

5.  If  2-thirds  of  a  barrel  of  flour  cost  $8,  what  will 
l-third  cost? 

6.  Eight  is  2  times  what  number? 

7.  If  3-fifths  of  a  pound  of  butter  cost  9  cents,  what 
will  1 -fifth  cost? 

8.  Nine  is  3  times  what  number  ? 

9.  If  4-fifths  of  a  pound  of  coffee  cost  16  cents,  what 
will  1 -fifth  cost? 

10.  Sixteen  is  4  times  what  number? 

11.  If  5 -sixths  of  a  gallon  of  wine  cost  35  cents,  what- 
will  1 -sixth  cost? 

12.  If  6-tenths  of  a  yard  of  '4oth  cost  30  cents,  what 
will  1 -tenth  cost? 

13.  If  4-sevenths  of  a  yard  of  muslin  cost  28  cents, 
what  will  1 -seventh  cost? 

14.  If  2-thirds   of  an  orange   cost  4  cents,  what  cost 
l-third?     If  l-third  cost  2  cents,  what  cost  the  whole? 

15.  Four  is  2-thirds  of  some  number:  what  is  l-third 
of  that  number?     2  is  l-third  of  what  number?     Then 
4  is  2-thirds  of  what  number  ? 

16.  If  2-thirds  of  a  yard  of  cloth  cost  $6,  what  cost 
1 -third  of  a  yard  ?    If  l-third  cost  $3,  what  eo3*-a  yard  ? 


72  KAY'S  INTELLECTUAL  ARITHMETIC. 

17.  Six  is  2-thirds  of  some  number:   what  is  1 -third 
of  that  number?'    3  is  1 -third  of  what  number?     Then 
6  is  2-thirds  of  what  number  ? 

18.  If  3-fourths  of  a  barrel  of  flour  cost  $6,  what  will 
1-fourth  cost?     If  1-fourth  cost  $2,  what  cost  a  barrel? 

19.  Six  is  3-fourths  of  some  number  :  what  is  1-fourth 
of  the   same  number?     2  is  1-fourth  of  what   number? 
Then  6  is  3-fourths  of  what  number  ? 

20.  If  2-fifths  of  a  melon  cost  8  cents,  what  cost  1-fifth? 
If  1-fifth  cost  4  cents,  what  cost  the  whole  ? 

21.  Eight  is  2-fifths  of  some  number  :  what  is  1-fifth 
of   the  same  number?     4   is    1-fifth   of  what  number? 
Then  8  is  2-fifths  of  what  number? 

22.  If  3-fifths  of  a  pound  cost  9  cents,  what  cost  1-fifth? 
If  1  -fifth  cost  3  cents,  what  cost  a  pound  ? 

23.  Nine  is  3-fifths  of  some  number :   what  is  1-fifth 
of  that  number?     3  is  1-fifth  of  what  number?     Then  9 
is  3-fifths  of  what  number  ? 

24.  If  4-fifths  of  a  pound  cost  8  cents,  what  cost  1-fifth? 
If  1-fifth  cost  2  cents,  what  cost  a  pound? 

25.  Eight  is  4-fifths  of  some  number:  what  is  1-fifth 
of  that  number?     2  is   1-fifth  of  what  number?     Then 

8  is  4-fifths  of  what  number  ? 

26.  If  3-fourths  of  a  dozen  eggs  cost   9  cents,  what 
fcost  1-fourth  ?     If  1-fourth  cost  3  cents,  what  cost  a  dozen  ? 

27.  Nine  is  3-fourths  of  some  number,  what  is  1-fourth 
of  that  number?     3  is  1-fourth  of  what  number?     Then 

9  is  3-fourths  of  what  number  ? 

28.  If  2-thirds  of  a  yard  of  tape  cost  20  cents,  what 
cost  1- third?     If  1 -third  cost  10  cents,  what  cost  a  yard? 

29.  Twenty   is    2-thirds    of    some    number :     what    is 
1 -third  of  that  number?     10  is  1 -third  of  what  number? 
Then  20  is  2-thirds  of  what  number  ? 

30.  Six  is  3-fourths  of  what  number? 


PARTS   OF   NUMBERS.  73 

ANALYSIS. — 1-fourth  is  l-third  of  3-fourths :  if  3-fourths 
are  6,  1-fourth  is  l-third  of  6,  which  is  2,  and  4-fourths3  (or 
the  whole  number,)  are  4  times  2,  which  are  8,  Am. 

31.  9  is     3-sovenths     of  what  number  ? 

32.  10  is  2-S3venths  of  what  number  ? 

33.  14  is  7 -eighths  of  what  number  ? 

34.  15  is  3-eighths  of   what  number  ? 

35.  16  IB  4-ninths  of   what  number  ? 

36.  18  is  6 -tenths  of  what  number  ? 

37.  20  is  5-fourths  of  what  number? 

38.  15  is  5-fourths  of  what  number? 
39  .  33  is  11 -sixths  of  what  number? 
40°.   22  is  2-elevenths  of  what  number? 
41°.  81  is  9-thirds  of  what  number? 


LESSON  II. 

1.  If  3-fourths  of  a  pound  of  raisins  cost  9  cents, 
how  much  will  a  pound  cost  ?  How  many  lemons,  at  2 
cents  each,  will  pay  for  1  pound  of  raisins  ? 

ANALYSIS.  —  First.  1-fourth  will  cost  l-third  as  much  as 
3-fourths,  and  4-fourths,  or  a  pound,  will  cost  4  times  as 
much  as  1-fourth  ;  if  3-fourths  cost  9  cents,  \-fourth  will  cost 
l-third  of  9  cents,  or  3  cents,  and  4-fourths  will  cost  4  times 
3  cents,  or  12  cents. 

Second.  At  2  cents  each,  it  will  require  as  many  lemons  as 
2  cents  are  contained  times  in  12  cents;  2  cents  are  contained 
in  12  cents  6  times.  Ans.  6  lemons. 


2.  If  2-thirds  of  a  pound  of  sugar  cost  16  cents, 
much  will  a  pound  cost  ?     How  many  oranges,  at  4  cents 
each,  will  pay  for  1  pound  of  sugar  ? 

3.  If   7-eighths   of  a  barrel   of  wine  cost  $42,  how 
many  bar.  of  cider,  at  $6  a  bar.,  will  pay  for  1  bar.  of  wine? 


74  KAY'S   INTELLECTUAL   ARITHMETIC. 

y 

4.  If  3-fifths   of  a  hogshead  of  sugar  cost  $24,  how 
many  barrels  of  flour,  at  $4  a  barrel,  will  pay  for  1  hogs- 
head of  sugar? 

5.  Sold  a  horse  for  §25,  which  was  5-eighths  of  his 
cost :  how  much  did  he  cost  me  ?     I  paid  for  him  with 
sloth,  at  $6  a  yard :  how  many  yards  did  I  give  ? 

6.  Thirty  is  5-eighths  of  how  many  times  5  ? 

Ans.^Of  as  many  times  5,  as  5  is  contained  times  in  the 
number  of  which  l-fifth  of  30  is  \-eiglitli. 

ANALYSIS. — If  30  is  5-eighths,  \-fiftli  of  30,  which  is  6,  is 
i-eighth,  and  the  number  is  8  times  6,  which  are  48;  48  is  9 
times  5  and  3-fifths  of  5. 

7.  12  is     4-sevenths  of  how  many  times  5? 

8.  18  is     3 -eighths     of  how  many  times  9  ? 

9.  16  is     2 -sevenths  of  how  many  times  9  ? 

10.  36  is  4-sevenths  of  how  many  times  8  ? 

11.  45  is  5 -ninths       of  how  many  times  7? 

12.  24  is  4-thirds       of  how  many  times  5  ? 

13.  72  is  8-fifths         of  how  many  times  7  ? 

14.  81  is  9-fourths     of  how  many  times  3  ? 

15.  50  is  10-sevenths  of  how  many  times  4? 

16.  63  is  7-sixths       of  how  many  times  5  ? 


LESSON  III. 

1.  James   gave  his   brother  4    marbles,    which   were 
2-thirds  of  all  he  had :  how  many  marbles  had  he  ? 

2.  Thomas    sold    a    knife    for    15   cents,    which    were 
3  fifths  of  its  cost :  how  much  did  it  cost  ? 

3.  William  lost  6  marbles,  which  were  3-eighths  of 
all  he  had  :  how  many  had  he  ? 

4.  I  sold  a  horse  for  S42,  which  were  6-fifths  of  hi? 
cost:  how  many  dollars  did  I  gain? 


PARTS   OF    NUMBERS.  75 

5.  A  grocer  sold  a  lot  of  flour  for  $40,  which  were 
5 -fourths  of  the  cost :  what  was  the  cost  ? 

6.  Sold  a  horse  for  $56,  which  were   8-fifths   of  the 
cost :  paid  with  flour,  at  $4  a  barrel :  how  many  barrels 
did  I  give? 

7.  A  man  sold  a  watch  for  $28,  which  were  4-thirds 
of  its  cost :  how  much  did  it  cost  ? 

8.  A  man  purchased  a  horse :  after  paying  3-fifths  of 
the  price,  he  owed  $20  :  what  was.  the  cost? 

9.  Alexander   sold    a    book    for    25    cents,    and  lost 
2-sevenths  of  the  cost :  what  was  the  cost  ? 

10.  If  a  boy  can  earn  32  cents  in  a  whole  day,  how 
\nuch  can  he  earn  in  5-eighths  of  a  day? 

11.  If  7-ninths  of  a  cask  of  wine  cost  $42,  how  much 
flour,  at  $8  a  barrel,  will  pay  for  1  cask  of  wine  ? 

12.  If  there  are  10  links  in  5 -ninths  of  a  chain,  how 
many  links  are  there  in  the  whole  chain? 

13.  In  an   orchard    there  are  12  cherry-trees :  the  re- 
maining 5-sevenths   of  the   orchard  are  apple-trees :    how 
many  apple-trees  are  there? 

14.  There  is  a  pole,  4-fifths  of  which  is  under  water, 
and  6  feet  out  of  water  :  how  long  is  the  pole  ? 

15.  There  is  a  pole,  3-fifths  of  which  is  in  the  earth, 
and  12  feet  in  the  air  :   how  long  is  the  pole  ? 

16.  One -fifth  of  a  pole  is  in  the  mud,  2 -fifths  in  the 
water,  and  14  feet  in  the  air :  how  long  is  the  pole  ? 

17.  The  age  of  Joseph  is  25  years,  which  is  5-eighths 
of  the  age  of  his  father :  the  father's  age  is  10  times  that 
of  his  youngest  son :  what  is  the  age  of  the  father?  whal 
is  the  age  of  his  youngest  son? 

18.  A  man  sold  a  horse  for  $45,  which  were  5 -thirds 
of  the  cost :  how  much  did  he  gain  ? 

19.  A   man   paid   $24  for   a   watch,    and    sold   it    for 
7-fourths  of  the  cost :  he  was  paid  in  cloth,  at  $5  a  yard : 
how  many  yards  did  he  receive? 


76  RAY'S   INTELLECTUAL   ARITHMETIC. 

20.  A  watchmaker  sold  a  watch  for  §18,  and  lost  2-fifths 
of  its  value  :  how  much  did  he  lose  ? 

ANALYSIS.— Since  he  lost  2-fifths,  he  sold  it  for  ^-fifths  of 
its  value;  then  ^-fifths  are  $18,  and  \-fifth  is  l-third  of 
$18,  or  $6;  2-fifths  are  2  times  $6,  or  $12,  Ans. 

21.  A  watchmaker  sold  a  watch  for  8-15,   and  gained 
2-sevenths  of  the  cost :  what  was  the  cost  ? 

22.  A  boy    spent   3-sevenths   of  his  money,  and    had 
12  cents  left :  how  much  had  he  ? 


SECTION  XII. 

LESSON  I. 

1.  If  you  have  12  cents,  and  3-fourths  of  your  money 
equal  l-third  of  mine,  how  many  cents  have  I  ? 

ANALYSIS. — \-fourth  of  12  is  3,  and  3-fourths  are  3  times 
3,  or  9 ;  9  is  l-third  of  3  times  9 :  3  times  9  are  27,  Ans. 

2.  2-thirds      of    9  are  1 -fifth       of  what  number? 

3.  4-fifths        of  10  are  1-half       of  what  number? 

4.  3-sevenths  of  14  are  1 -sixth    -of  what  number? 

5.  5-sixths      of  12  are  1-fourth   of  what  number? 

6.  3-eighths    of  16  are  1 -fifth       of  what  number? 

7.  2-fifths        of  20  are  how  many  thirds      of  15  ? 

8.  4-sevenths  of  21  are  how  many  fifths        of  25  ? 

9.  5-sixths      of  30  are  how  many  fourths    of  32  ? , 

10.  7 -ninths      of  45  are  how  many  sevenths  of  21  ? 

11.  3-fourths    of  36  are  how  many  thirds      of  18  ? 

12.  5-eigl^ths    of  40  are  how  many  ninths      of  36  ? 

13.  Divide   1-fourth  of  32  by  2-thirds   of  9.     Three- 
fourths  of  28  by  2-thirds  of  12. 


PARTS   OF    NUMBERS.  77 

14.  Divide  3-fifths   of  40  by  3-sevenths  of  14.     Five* 
sevenths  of  42  by  3-eighths  of  24. 

15.  Divide   the    7-ninths    of    54    by   5-sixths    of   12. 

16.  Nine-elevenths  of  88  are  how  many  times  5-ninths 
of  18? 


LESSON  II. 

1.  William  says  to  Frank,  "  Your  age  is  15  years, 
and  4-fifths  of  your  age  are  6  -sevenths  of  mine  :  what 
is  my  age  ?" 

ANALYSIS.  —  l-fifth  of  15  is  3,  and  4-fifths  are  4  times  3, 
which  are  12;  if  12  is  Q-sevenths,  l-sixth  of  12,  which  is  2, 
is  l-seventh,  and  if  2  is  l-seventh,  the  number  is  7  times  2, 
which  are  14.  Ans.  14  years. 


2. 

3-fourths    of 

8 

are 

2-thirds 

of  what  number  ? 

3. 

4-fifths        of 

20 

are 

8-ninths 

of  what  number  ? 

4. 

3-fifths       of 

20 

are 

4-fifths 

of  what  number? 

5. 

5-sixths      of 

36 

are 

10-thirds 

of  what  number  ? 

6. 

6  -sevenths  of 

28 

are 

3-fifths 

of  what  number? 

7. 

&-  eighths    of 

32 

are 

4-sevenths 

of  what  number  ? 

8. 

8-ninths     of 

45 

are 

10-elevenths 

of  what  number  ? 

9. 

Five-sevenths 

of 

21 

are  3-fourths 

of  5  times  what  ? 

10. 

Four-ninths  of  36  are  8-sevenths 

of  7  times  what  ? 

11. 

Three-tenths 

of  40 

are  6-fifths  of  10  times  what? 

12. 

Seven-eighths 

of 

64 

are  7-ninths  of  12  times  what? 

13. 

Three  -fourths 

of 

16    are    2-thirds    of  how    many 

times  1  -fifth  of  20? 

ANALYSIS.  —  \-Jourih  of  16  is  4,  and  3-fourths  are  3  times 
4,  which  are  12  ;  if  12  is  2-thirds,  I  -half  of  12,  which  is  6, 
is  1-third,  and  6  is  l-third  of  3  times  6,  ivJiich  are  18  ;  l-Jifth 
of  20  is  4,  and  18  is  4  times  4  and  2-fourths  of  4.  Ans.  4 
and  2-fourths  times. 


78  RAY'S   INTELLECTUAL   ARITHMETIC. 

14.  Four-fifths   of   20   are    8-sevenths   of  how    many 
times  1 -third  of  15  ? 

15.  Six-sevenths    of    21    are    2-fifths    of    how    many 
times  1 -fifth  of  30? 

16.  Four-ninths    of    27    are    2-thirds    of   how    many 
times  1 -seventh  of  35  ? 

17.  Five-sevenths   of  42    are   5 -ninths   of  how   many 
times  1 -eighth  of  56  ? 

18.  Three-fourths    of   24  are   9-tenths   of   how  many 
times  2-fifths  of  10  ? 

19.  Five-sixths    of   12   are   2-sevenths    of  how  many 
times  3-fourths  of  8  ? 

20.  Five-eighths    of    32    are    4-ninths   of   how   many 
times  that  number  of  which  2  is  1 -third? 

21.  Six-sevenths  of  28   are    8-elevenths  of  how  many 
times  that  number  of  which  4  is  2-fifths  ? 

22.  Eight-ninths  of  45   are     10-thirds   of  how   many 
times  that  number  of  which  14  is  7-fourths? 

23.  Seven-tenths    of   50    are    5-ninths    of    how    many 
times  that  number  of  which  8  is  4-fifths  ? 

24.  Nine -sevenths  of  56   are    12 -fifths   of  how  many 
times  the  number  of  which  6  is  6 -sevenths  ? 

25.  One-third    of  a    certain    number   is  2   more   than 
1-half  of  12  :  what  is  the  number  ? 

26.  One-fourth   of    a   certain    number   is    3   less   than 
1 -fifth  of  30:  what  is  the  number? 

27.  Two-fifths  of  20  is  6  less  than  how  many  thirds 
of  21  ? 

28.  Three-fourths  of  24  is   6   more  than   2-thirds  of 
what  number  ? 

29.  Five-sixths   of  30,  increased   by  4,  is  1   less  than 
3-fourths  of  some  number :  what  is  that  number  ? 

30.  Three -fifths  of  40  is  just  3  less  than  9 -sevenths  of 
how  many  times  7  ? 


MISCELLANEOUS    REVIEW.  79 


SECTION  XIII.— REVIEW. 

LESSON  I. 

1  William  had  23  cents :  Thomas  gave  him  8  cents 
more,  treorge  6,  James  5,  and  David  7:  he  gave  15  cents 
for  a  book:  how  many  cents  had  he  left? 

2.  A  grocer  paid  $12  for  sugar,  $9  for  coffee,  $5  for 
tea,  $7  for  flour,  and  had  §10  left :  how  many  dollars  had 
he  ai  first? 

3.  A  boy  had  11  cents :  his  father  gives  him  9  cents, 
his  mother  6,  and  his  sister  enough  to  make  34 :    how 
many  cents  did  his  sister  give  him  ? 

4.  Five  men  bought  a  horse  for  $42 :   the  first  gave 
$13,  the  second  7,   the   third  5,  and  the  fourth  9:  how 
many  dollars  did  the  fifth  give  ? 

5.  A  man  purchased  8  sheep  at  $4  a  head,  5  barrels 
of  flour  at  $3  a  barrel,  4  yards  of  cloth   at  $3  a  yard, 
and  5  ounces  of  opium  at  $1  an  ounce :  how  much  did 
he  spend? 

6.  A  boy  lost  25  cents :  after  finding  15  cents,  he  had 
25  cents :  how  many  cents  had  he  at  first  ? 

7.  A  man  owed  a  debt  of  $28,  and  paid  all  but  $9 : 
how  much  did  he  pay? 

8.  Borrowed  $56 :  at  one  time  paid  $23 ;  at  another, 
all  but  $7 :  how  much  did  I  pay  the  last  time  ? 

9.  James  borrowed  37  cents :  at  one  time  he  paid  5, 
at  another  8,  and  the  third  time,  all  but  15.:  how  many 
cents  did  he  pay  the  third  time  ? 

10.  A  farmer  sold  2  cows  at  $9  each,  and  5  hogs  at 
$3  each,  and  received  in  payment  3  sheep  at  $3  each,  and 
the  rest  in  money :  how  much  money  did  he  receive  ? 

11.  A  farmer  sold  12  barrels  of  cider  at  $3  a  barrel : 
he  then  purchased  5  barrels  of  salt  at  $3  a  barrel,  and 
some  sugar,  for  $8 :  how  many  $'s  had  he  left? 


80  BAY'S   INTELLECTUAL   ARITHMETIC. 

12.  A  merchant  purchased  13  hats  at  §4  each,  5  pairs 
of  shoes  at  §2    a   pair,   and   an  umbrella,  for  §7  :  what 
must  he  sell  the  whole  for,  to  gain  $9  ? 

13.  If  1  barrel  of  flour  cost  S3,  what  cost  7  barrels? 

14.  If  2  pounds  cost  16  cents,  what  cost  5  pounds? 

15.  If  3  barrels  of  cider  cost  §12,  what  cost  4  barrels  ? 

16.  If  4  yards  of  cloth  cost  $28,  what  cost  7  yards  ? 

17.  If  3  pounds  cost  27  cents,  what  cost  8  pounds  ? 

18.  If  5  barrels  of  flour  cost  §35,  what  cost  8  barrels  ? 

19.  If  7  apples  cost  28  cents,  what  cost  3  apples? 

20.  If  8  oranges  are  worth  24  apples,  how  many  apples 
are  3  oranges  worth  ? 

21.  If  4  pounds  of  cheese  cost  36  cents,  what  will  3 
pounds  cost? 

22.  If  8  yards  of  cloth  cost  $56,  what  cost  7  yards  ? 

23.  What  sum  of  money  must  be  divided  among  11  men 
that  each  shall  receive  §9  ? 

24.  A  walks   5   miles,  while  B  walks  3 :  when  A  ha\ 
gone  35  miles,  how  far  has  B  gone  ? 

ANALYSIS. — In  35  there  are  7  fives,  and  B  walks  as  many 
threes  as  A  walks  fives;  that  is,  7  times  3,  or  21  miles. 

25.  Joseph  and  his  father  are  husking  corn :  the  father 
can  husk  7  rows  while  Joseph  husks  3 :  how  many  rows 
will  Joseph  husk  while  his  father  husko  42  ? 

26.  Charles  can   earn   §9  while   Mary  earns  §4 :    how 
many  S's  will  Charles  earn  while  Mary  earns  28  ? 

27.  If  6  horses  eat  12  bushels  of  oats  in  a  week,  how 
many  will  10  horses  eat  in  the  same  time  ? 

28.  If  5  horses  eat  16  bushels  in  2  weeks,  how  long 
would  it  take  them  to  eat  56  bushels  ? 

29.  If  6  apples  are  worth  18  cents,  how  many  apples 
must  be  given  for  5  oranges  worth  6  cents  each  ? 

30.  How  many  horses   can  eat   in  9   days,  the   same 
amount  of  hay  that  12  horses  eat  in  6  days? 


MISCELLANEOUS    REVIEW.  81 

31°.  If  5  men  earn  $30  in  3  days,  how  much  will  2 
men  earn  in  the  same  time  ?  How  much  will  2  men  earn 
in  I  day? 

LESSON   II. 

1.  How  many  times  6  in  3-fifths  of  40? 

2.  How  many  times  3-sevenths  of  14  in  54  ? 

3.  How  many  times  5-sixths  of  12,  in  4-ninths  of  72? 

4.  How  many  times  3-fifths  of  20,  in  twice  that  num- 
ber of  which  14  is  7-ninths  ? 

5.  If  3-eighths  of   a  tun  of  hay  cost  $9,  what  will 
5-sixths  of  a  tun  cost  ? 

6.  If  a  man  can  earn  $7  in  1 -fifth  of  a  month,  how 
many  dollars  can  he  earn  in  1  month  ? 

7.  If  a   man   can   earn  $42   in   1  month,  how  many 
dollars  can  he  earn  in  1 -sixth  of  a  month? 

8.  If  a  hogshead   of   sugar  is  worth  $96,  what  are 
7-eighths  worth? 

9.  Five  men  paid  $20  for  a  horse :  what  part  of  $20 
did  2  men  pay  ? 

10.  If  $7   will  buy  56   yards   of  muslin,  how  many 
yards  will  $4  buy  ? 

11.  If  3  men  can  do  a  job  of  work  in  16  days,  in  how 
many  days  can  4  men  do  it  ? 

12.  If  3  men  spend  $12  in  1  week,  at  the  same,  rate, 
how  many  would  2  men  spend  in  6  weeks  ? 

13.  If  6  men  do  a  piece  of  work   in   7  days,  in  how 
many  days  can  3  men  do  it  ? 

14°.  If  5  men  do  a  piece  of  work  in  8  days,  in  how 
many  days  can  4  men  do  a  job  twice  as  large  ? 

15°.  If  6  men  do  a  job  in  5  days,  in  how  many  days 
can  2  men  do  a  job  half  as  large? 

16.  James  had  16  apples :  he  kept  1-fourth  himself 
and  divided  the  remainder  equally  among  3  of  his  com- 
panions :  how  many  did  each  receive  ? 


82  RAY'S   INTELLECTUAL  ARITHMETIC. 

-**•; 

17.  Three-fourths  of  24,  increased   by  2-thirds  of  12, 
are  equal  to  how  many  ? 

18.  Five-sixths  of  24,  diminished  by  3-fourths  of  20, 
equal  how  many  ? 

19.  One-half,  and   2-thirds,   and  3-fourths  of  12    are 
how  many  ? 

20.  Two-thirds  of  12,  less  1-half  of  12,  are  2-fifths  of 
what  number? 

21°. From  10  take  3-fifths  of  itself;  add  to  the  remain- 
der its  1-half:  what  is  the  result? 

22.  Thomas  had  28  cents :    he  gave*  2-sevenths  to  his 
sister,  and  2-fifths  of  the  remainder  to  his  brother :  how 
many  more  did  he  give,  than  he  had  left? 

23.  James    had    35    marbles :     he    gave    to    Thomas 
3-sevenths,  to  Charles  2-fifths :  to  which  did  he  give  the 
most,  and  how  many  ?     What  number  had  he  left  ? 

24.  Thomas  had  §28  :  he  kept  2-sevenths,  and  divided 
the  remainder  equally  among  his  4  brothers:  how  many 
dollars  did  each  receive  ? 

25.  A  grocer  had  14  barrels  of   flour :  he  sold  4-sev- 
enths  at  S3  a  barrel,  and  the  remainder  at  §5  a  barrel : 
what  did  it  amount  to  ? 

26.  Bought  15  yards   of  cloth,   at  82  a  yard  :   I  sold 
1 -third  at  §4  a  yard,  2-fifths  at  83   a  yard,  and  the  re, 
mainder  at  85  a  yard :  how  much  did  I  gain  ? 

27.  Bought  10  yards  of  cloth  for  §90,  and  sold  2-fifths 
of  it  for  840  :  how  much  a  yard  did  I  gain  ? 

28.  Two  men  travel  the  same  direction :  A  is  40  miles 
ahead  of  B  ;  but  B  travels   23  miles  a  day,  and  A  18 : 
in  how  many  days  will  B  overtake  A  ? 

29.  A   hare   is  90  yards  in  advance  of  a  hound  :  the 
hound  goes  10  yards  in  a  minute,  and  the  hare  7 :  in  how 
many  minutes  will  the  hound  overtake  the  hare?     How 
far  will  each  run  ? 

30  .If  a  hound  runs  7  rods  while  a  hare  runs  4, 
far  will  the  hare  run  while  the  hound  runs  35  rods  ? 


MISCELLANEOUS    KEVIEW.  83 

31.  One  man  is  pursuing  another,  who  is  4  miles  in 
advance  :  he  travels  3  miles  in  pursuit,  while  the  other 
advances  1  :  how  much  aoes  he  gain  in  going  3  miles  ? 
How  far  must  he  travel  to  overtake  him  ? 

32°.  C  and  D  travel  in  tne  same  direction :  C  is  15  miles 
ahead  of  D ;  but  D  travels  5  miles  an  hour,  and  C  only 
2 :  in  how  many  hours  wnl  L  overtake  C  ?  How  far  will 
D  have  traveled  ? 

33.  A  cistern  containing  2±  gallons,  is  filled  by  a  pipe 
at  the  rate  of  8  gallons  an  hour,  and  emptied  by  a  pipe 
at  the  rate  of  5  gallons  an  hour :  if  both  pipes  are  open, 
how  many  gallons  will  remain  in  the  cistern  each  hour? 
How  long  will  the  cistern  be  in  filling  ? 

34  .A  cistern  containing  36  gallons  has  2  pipes ;  by 
the  first  it  receives  6  gallons  an  hour,  and  by  the  second 
it  discharges  9  gallons  an  hour :  if  both  pipes  are  left 
open,  how  long  will  it  take  to  empty  the  cistern  ? 

35.  My  pants  cost  $8,  which  were  2-fifths  the  cost  of 
my  coat ;  my  vest  cost  1-haif  as  much  as  my  pants :  what 
was  the  cost  of  the  whole  ? 


LESSON   III. 

1.  What  are  the  divlbors  of  a  number  called  ?     Ans. 
Its  factors. 

2.  What   is    the    divisor  of   a   number  ?     Ans.    Any 
number  which  is  contained  in  it  an  exact  number  of  times; 
that  is,  without  a  remainder.  • 

3.  What  is   a   multiple    of   a   number  ?      Ans.    Any 
product  of  which  that  number  is  a  factor. 

4.  Twelve  can  be  divided  by  2,  3,  4,  and  6 :  what  are 
the  factors  of  12  ?     of  24  ? 

5.  What   2   numbers   multiplied  together,  will  make 
28  ?     What  are  the  factors  of  28  ? 

6.  What   numbers   w'll  divide   22?      What    are   th* 
factors  of  22  ? 


84  RAY'S   INTELLECTUAL   ARITHMETIC. 

7.  Six  is  a  factor  of  24;  what  number  is  a  multipk 
of  6  ?     of  12  ? 

8.  Seven  is  a  factor  of  21  ?     What  number  is  a  mul- 
tiple of  7  ?     of  3  ? 

9.  What  three   numbers  are  multiples  of  7  ?     of  8  ? 
of  9?     of  10? 

10.  What  numbers  will  divide  18?     27?     33?     36? 
39?    48?    49?     63?     64?     65?     72?    75? 

11.  What  number   will    divide    both  10  and  15?     12 
and  16  ?     20  and  32  ?     64  and  80  ?     72  and  96  ? 

12.  What  is  the  greatest  number  less  than  12,  that  will 
divide  12  ?     Less  than  16  that  will  divide  16  ?     Less  than 
20  that  will  divide  20  ? 

13.  What  i?  the  greatest  number  that  will  divide  both 
6  and  10  ?     12  and  18  ?     20  and  32  ?     24  and  36  ?     42 
and  48  ?     56  and  72  ? 

14.  What   numbers   can   be   divided  by  9  ?     12  ?     8  ? 
7?     16?     14?     15? 

15.  What  is  the  least  number  that  can  be  divided  by 
2  and  3?    4  and  7  ?     2  and  9  ?     7  and  21  ?     10  and  12? 
12  and  20?  * 

The  answers  to  the  13th  question  are  called  the  Greatest 
Common  Divisors ;  to  the  15th  question,  the  Least  Common 
Multiples  or  Dividends. 


SECTION  XIV.— FRACTIONS. 

LESSON  I. 

A  single  or  whole  thing  of  any  kind,  is  called  a  unit 
one ;  as  1  yard,  1  dollar,  1  mile,  1,  &c. 


1.  The   above   line   represents   a  yard  of  tape  :  it  is 
called  1  yard. 


FRACTIONS.  85 

2.  If  you  divide  it  into   two  equal  parts,  one  of  the 
$arts  is  called  one-half  of  a  yard. 

one-half.  one-half. 

One  part  is  represented  thus,  |,  and  is  read  one-half. 

3.  If  the   yard   of  tape   is  divided   into    three  equal 
parts,  one  of  the  parts  is  called  one-third;  two  parts  are 
called  two-thirds. 

one-third.  one-third.  one-third. 

One  part  is  represented  thus,  ^,  and  is  read  one-third; 

2  parts  are  represented  thus,  |,  read  two-thirds. 

4.  If  the  yard  of  tape  is  divided  into  four  equal  parts, 
one  of  the  parts  is  called  one-fourth;  two  parts  two-fourths; 

3  parts  three-fourths. 

one-fourth.      one-fourth.       one-fourth.      one-fourth. 

One  part  is  represented  thus,  |,  and  is  read  one-fourth ; 
2  parts  are  represented  thus,  |,  read  two-fourths;  3  parts 
thus,  |,  read  three-fourths. 

5.  If  the  yard  of  tape  is  divided  into  five  equal  parts, 
one  of  the  parts  is  called  one-fifth;  two  parts  two-fifths; 
three  parts  three-fifths. 


A  FRACTION  is  a  part  of  a  unit,  or  one.     In  every 
fraction  the  unit  is  divided  into  equal  parts.     The  lower 
number   shows   into   how   many   equal    parts  the  unit  is 
divided  :  the  upper  number,  how  many  parts  are  tak^n, 
2d  Bk.          6 


86  RATS   INTELLECTUAL   ARITHMETIC. 

The  lower  number  is  called  the  DENOMINATOR,  because 
it  denominates  or  names  the  parts  :  the  upper  number  the 
NUMERATOR,  because  it  numbers  the  parts. 

In   reading  fractions,   that   is,  in   expressing  them  bj  ' 
words,  read  the  Numerator  first  j  the  Denominator  last. 

6.  If   you    divide    a   yard  into    6    equal    parts,  whai 
figures  will  express  2  parts  ?     What  5  parts  ? 

7.  If  you  take  away  2  of  the  6  parts,  what  fraction 
will  express  the  remainder  ? 

8.  How  can  you  express  in  figures  3-sevenths  of  1  ? 

9.  In  the  fraction  f ,  into  how  many  parts  is  the  unit 
(one)  divided,  and  how  many  are  represented  ? 

10.  Read  the  following  Fractions: 
11111111       i       i       i       i       i       i       i 

~g>    3>    o?    6>    7)    6'    ¥?   To>    TT>    72>    7.3?    74>    To?    76?    T7> 

i  I,  I,  I.  4,  I  i ,  T3«j>  T9T,  IS.  T9a>  if,  M>  &  IS- 

11.  What   does    the   fraction  f   represent?     Into  how 
many  equal  parts  is  the  unit  (one)  divided? 

12.  What  do  you  understand  by  T9T  ?     j-f  ?     |f  ? 

13.  Divide  an  apple  into  10  equal  parts,  and  take  away 
3  parts  :  what  fraction  will  express  the  remainder  ? 

14.  What  fraction  will  represent  1-half  of  1  ?     1-half 
of  3  ?     1-third  of  5  ? 

15.  Which   is  the  greater  fraction,  4  or  4  ?     -  or  j  ? 
ior£?     |or|? 

16.  What  is  a  fraction  ?     What  does  the  denominator 
of  a  fraction  show  ?     What  does  the  numerator  show  ? 

17.  How  many  apples  in   2-halves    of  an  apple  ?     In 
3-thirds  ?     In  4-fourths  ?     In  5 -fifths  ? 

18.  When  the  numerator  of  a  fraction  is  equal  to  the 
denominator,  as  |,  |,  |,  &c.,  what  is  its  value  ?     Ans.  1. 

19.  Are  3-halves  of  an  apple  more  or  less  than  a  whole 
apple  ?     When  the  numerator  is  greater  than  the  denom- 
inator, is  the  value  of  the  fraction  greater  or  less  than  1  ? 


FKACTIONS,  £7 

20.  If   jker  denominators  of  fractions  be  increased,  do 
the  fractions  become  greater  or  less  ?     Why  ? 

21.  If   the   numerators   be  increased,  do  the  fractions 
become  greater  or  less  ?     Why  ? 

22.  Such  fractions  as   §,  |,  |,  &c.,  being  either  equal 
to,  or  greater  than  1,  are  called  improper  fractions;  those 
less  than  1,  as  ^,  |,  &c.,  proper  fractions. 

23.  Whole   numbers  and  fractions  joined  together,  as 
14,  2|,  5j,  6|,  10|,  &c.,  are  called  mixed  numbers. 


SECTION  XV. 

LESSON   I. 

1.  How  do   you   find   the   number   of  halves  in  any 
number  ?     Why  ? 

2.  In  1,  how  many  halves  ?     In  U  ?     In  2  ?     In  2^  ? 
In  3  ?     In  3J  ?     In  4  ?     4 '  ?      5  ?      6  ?     7  ?     7i  ? 

3.  How  do  you  find  how  many  thirds  there   are   in 
any  number  ?     Why  ? 

4.  How  many  thirds  in  1  ?     In  1|  ?     In  If  ?     In  2  ? 
In  21?    In  23?    3?    31?    3§  ?     4?     4 4  ?     5?     6? 

O  O  DO  O 

5.  How  do  you  find  how  many  fourths  there  are  in 
any  number  ?     Why  ? 

6.  How  many  fourths  in  1  apple?    In  1  and  1-fourth? 
In  1  and  2-fourths  ?     In  1  and  3-fourths  ?     In  2  ?     In  2 
and  3-fourths  ?     In  3  ?     In  4  and  3-fourths  ? 

7.  How  find  how  many  fifths  are  in  any  number  ? 

8.  How  many  fifths  in  1  ?    In  H  ?    In  If?     In  If  ? 
In  11  ?     In  2  ?      In  2i  ?      In  2|  ?     In  2|  ?    In  2|  ?  '  IB 
B?    "inSf?     In3£?°In4f?      In  54? 

9.  In  4  and  2-sevenths  how  many  sevenths  ? 
10.    In  5  and  3 -eighths    how  many  eighths  ? 
Jl.   In  4  and  7-ninths      how  many  ninths? 


88  RAY'S    INTELLECTUAL    ARITHMETIC 

12.  In  5  and  3-tenths  how  many  tenths  ? 

13-  In  12  and  1-third  how  many  thirds? 

14  In  11  and  1-half  how  many  halves  ? 

15-  In  6  and  3-fifths  how  many  fifths  ? 

16*  In  7  and  3-fourths  how  many  fourths? 

17.  In  6  and  1-sixth  ^  how  many  sixths  ? 

18.  In  5  and  7-eighths  how  many  eighths  ? 

19.  In  11  and  4-sevenths  how  many  sevenths  ?J 

20.  In  7  and  9-tenths  how  many  tenths? 

LESSON   II. 

1.  James's  brother   gave   him  3-halves  of  an  apple: 
how  many  apples  did  he  give  him  ?      Ans.    One  apple 
and  1-half. 

2.  In  3  halves  how  many  ones  ? 

ANAL. — Since  2-halves  make  1,  in  3-halves  are  as  many  ones 
as  2-halves  are  contained  times  in  3-halves;  2-halves  in 
3-halves,  one  and  one-half  times.  Ans.  1  and  1-half. 

3.  John's  father  gave  him  4  half  -  dollars :  how  many 
dollars  did  he  give  him  ? 

4.  How   many   ones   in   4-halves?     in  5-halves?     in 
6-halves  ?     in  7 -halves  ?     in  8-halves  ?     in  9-halves  ? 

5.  How  do  you  find  how  many  ones  there  are  in  any 
number  of  halves? 

6.  If  you  divide  an  orange  into  3  equal  parts,  what  is 
1  part  called  ?     How  many  thirds  in  1  orange  ? 

7.  If  1  orange  make  3-thirds,  how  many  oranges  would 
there  be  in  3-thirds   of  an  orange?     in  4-thirds  of  an 
orange?     in  5 -thirds?     in  6 -thirds? 

8.  How    many   ones   in   3-thirds?     in   4-thirds?     in 
5-thirds  ?     in  6-thirds  ?     in  7- thirds  ?     in  8-thirds  ? 

9.  How  do  you  find  how  many  ones  there  are  in  any 
number  of  thirds  ? 


FRACTIONS.  89 

10.  In    4-fourths  of  a  dollar  how  many  dollars  ? 
In    5-fourths?  In    6-fourths  ? 
In    7-fourths?  In    8-fourths  ? 
In    9-fourths?  In  11-fourths? 
In  13-fourths  ?                         In  15-fourths  ? 
In  18-fourths  ?                         In  19-fourths  ? 

11.  How  many  ones  in  4-fourths  ?     in  5-fourths  ?    in 
6-fourths?    in  7-fourths?     in  8  fourths?     in  9-fourths? 
11-fourths?     13-fourths?     15-fourths?     19-fourths? 

12.  How  do  you  find  how  many  ones  there  are  in  any 
number  of  fourths  ? 

13.  In    5 -fifths  of  an  orange,  how  many  oranges  ? 
In    6-fifths?  In    7-fifths? 

In    8-fifths?  In    9-fifths? 

In  10-fifths  ?  In  11-fifths  ? 

Inl3-fifths?  Inl5-fifths? 

Inl7-fifths?  In  18-fifths? 

14.  How  many  ones  in  5-fifths  ?     6-fifths  ?     7-fifths  ? 
8-fifths?    9-fifths?     10-fifths?     11-fifths?    13-fifths? 

15.  How  do  you  find  how  many  times  1  there  are  in 
any  number  of  fifths  ? 

16.  How  many  times  1,  that  is,  how  many  whole  ones, 
in  23 -sixths  ?     Ans.   Three  and  5 -sixths. 

17.  In  30-sevenths  are  how  many  times  1  ? 

18.  In  35-sevenths  ?  22.   In  23-halves  ? 

19.  In46-ninths?  23.    In  3 3 -fifth s  ? 

20.  In  53-tenths  ?  24.    In  31-fourths  ? 

21.  In37-thirds?  25.   In37-sixths? 

26.  In  47-eighths  are  how  many  whole  ones  ? 

27.  In  81-sevenths  ?  30.    In75-tenths? 

28.  In79-tenths?  31.    In  89-elevenths? 

29.  In  53-ninths  ?  32.    In  93-twelfths  ? 


90 


KAY'S   INTELLECTUAL   ARITHMETIC. 


SECTION    XVI. 

LESSON  I. 

ILLUSTRATION. — The  first  line  represents  a  yard  of 
tape  divided  into  two  equal  parts ;  the  second,  a  yard  of 
tape  divided  into  four  equal  parts. 

one-half.  one-lialf. 


one-fourth.        one-fourth.        one-fourth.       one-fourth. 
One-half  Q),  is  equal  to  two-fourths  (f). 

1.  If  I  give  to  Mary  1-half  of  an  orange,  and  to  Jane 
1 -fourth,  how   much  more  will   Mary  have  than  Jane? 
what  part  will  be  left  ? 

2.  How  much  greater  is  1-half  than  1 -fourth  ?     How 
much  are  1-half  and  1 -fourth? 

3.  James  divided  a  melon,  giving  to  his  sister  1-half, 
and  to  his  brother  1 -fourth  :  what  part  did  he  give  away  ? 

4.  Thomas  gave  3-fourths  of  a  dollar  for  a  geography, 
and  1-half  of  a  dollar  for  an  arithmetic  and  slate :  how 
much  did  he  give  for  both  ? 

5.  How  much  are  1-half  and  3-fourths? 

ILLUSTRATION. — The  first  line  represents  a  yard  of 
ribbon  divided  into  2  equal  parts;  the  second,  a  yard 
divided  into  six  equal  parts. 

one-half.  one-half. 


I -sixth.      1  -sixth.      1- sixth.     I-sixth.     1-sixth.     1-sixth. 

6.  One-half    is   how    many  sixths?     1-third   is  how 
many  sixths  ?     2-thirds  are  how  many  sixths  ? 


FRACTIONS.  91 

7.  James  received  1-half  of  an  orange,  and  Charles 
I-third :  how  much  more  had  James  than  Charles  ? 

J3.  How  much  are  1-half  and  1 -third? 

9.  If  1 -third  is  2-sixths,  how  many  sixths  are  there 
in  2-thirds  ? 

10.  A  yard  of  flannel  costs  half  a  dollar :   a  yard  of 
sloth  2-thirds  of  a  dollar :  how  much  do  both  cost  ? 

11.  James  bought  2  melons;  he  gave  to  Lucy  half  of 
the  first ;  to  Jane  2-thirds  of  the  second  :  what  part  of  a 
melon  had  Jane  more  than  Lucy  ? 

12.  How  much  greater  are  2-thirds  than  1-half? 

13.  After  taking  away  1-half  and  1 -third  of  an  apple, 
what  part  will  be  left  ? 

14.  I  wish  to  divide  an  orange,  and  give  to  Mary  1-half, 
to  Jane  1 -fourth,  and  to  William  1-eighth:  how  must  1 
divide  it  ?     how  many  eighths  will  each  have  ? 

15.  Two -fourths  are  how  many  eighths  ?     3 -fourths  are 
liow  many  eighths  ? 

16.  One-fifth  is  how  many  tenths?     2-fifths  are  how 
many  tenths  ?     3-fifths  ?     4-fifths  ? 

17.  Thomas  wishes  to  divide  an  orange,  and  give  Ann 
1-half,  and  Lucy  2-fifths:  how  must  he  divide  it?  what 
part  will  he  have  left  ? 

18.  How  much  are  1-half  and  2-fifths? 

19.  One-third  is  how  many  twelfths?     1-fourth  is  how 
many   twelfths?     How   many   twelfths   in   2-thirds?     in 
2-fourths  ?     in  3-fourths  ? 

20.  A  farmer  sows  1-half  of  a  field  in  wheat,  1-third 
in  rye ;  the  rest  in  barley :    how  many  twelfths  are  in 
wheat  ?     how  many  in  rye  ?     how  many  in  barley  ? 

21.  How  many  twelfths  in  1-fourth?    in  1-fourth  and 
1-third? 

22.  David  bought  a  pound  of  figs:  he  gave  1-third  of 
them  to  his  mother,  1-fourth  to  his  sister,  and  1 -sixth  to 
his  brother :  what  part  had  he  left  ? 


2  RAY'S   INTELLECTUAL   ARITHMETIC. 

23.  How  many  eighths  are  in  1-half?     in  3-fourths? 

24.  How    many   tenths    are    in    1-half?      in   1 -fifth? 
in2-fifths?     in  3 -fifths  ? 

25.  How    many    fifteenths    in    1-third?      in    1-fifth? 
in  3-fifths  ?     in  4-fifths  ? 

26.  How   many   twentieths   in    1-half?      in   1 -fourth  ? 
in  1-fifth  ?     in  3-fourths  ?     in  3-fifths  ? 

27.  Reduce  -f  and  f  to  twelfths :  ^  and  f  to  fifteenths : 
also,  i  and  |  to  tenths. 

28.  Reduce  f  and  f  to  twentieths  :     ^,  |,  and  1-  to  twen- 
tieths. 

2&  Reduce  |  and  ^  to  fourteenths:  ^  and  f  to  eight- 
eenths. 

30.  Reduce  f,  f,  and  |  to  twenty-fourths :  |  and  T30  to 
fortieths. 

REMARK. — When  two  or  more  fractions  have  the  same  de- 
nominator, they  are  said  to  have  a  common  denominator; 
thus,  |  and  |  have  6  for  a  common  denominator. 

When  the  denominators  of  two  fractions  are  not  the  same,  a 
common  denominator  may  be  found  by  multiplying  the  denomi- 
nators together. 

•     Find  the  Common  Denominator 


31.      Of   £  and  f  . 

1  and  f 

1  and  |. 

32.      Of  |  and  f  . 

1  an<*  f  • 

^  and  jl. 

33.      Of  |  and  |. 

|  and  |. 

i  and  4. 

34.      Of   f  and  f  . 

|  and  f  . 

|  and  f  . 

35,      Of   f  and  f 

i  and  f  . 

•|  and  |. 

36.      Of  |  and  Jfc. 

-|and/T. 

§  and  ^ 

37.      Of   1  and  f 

|  and  y^. 

1  and  |. 

38.      Of  |  and  |. 

1  and  |. 

1  and  |. 

39°.Reduce  i,  |,  f, 

and  j|  to  thirty-seconds. 

40°.Reduce  4,  |,  U 

,  and  4|  to  fif 

fcy-fourtha. 

FRACTIONS. 


93 


41°.Reduce  |,  |, 
42°.  Reduce  -J,  ^, 


j,  T73,  and  ^  to  forty-eighths. 

J,  ^,  £,  TV,  and  y^  to  seventy-seconds, 


LESSON  II. 

ILLUSTRATION.  —  The  first  line  represents  a  yard  of 
ribbon  divided  into  2  equal  parts  ;  the  second  line,  a 
yard  divided  into  4  equal  parts. 

one-half.  one-half. 


one-fourth.        one-fourth.        one-fourth.        one-fourth. 


The  first  of  the  lines  below  represents  a  yard  divided 
into  3  equal  parts ;  the  second,  a  yard  divided  into  6  equal 
parts.  From  this  it  is  seen  that  ^  is  equal  to  f ;  and 
that  |  are  equal  to  |. 

one-third.  one-third.  one-third. 


1-sixth.     I-sixth.     \-sixth.     1-sixth.     1-sixth.     1-sixth. 

REMARK. — The  Numerator  and  Denominator  are  called  the 
terms  of  the  fraction;  when  these  are  the  smallest  numbers  of 
-which  the  fraction  admits,  it  is  said  to  be  in  its  lowest  terms. 

1.  Reduce  §  to  its  lowest  terms.     Divide  loth  terms  ly 
the  greatest  number  (3),  that  will  exactly  divide  loth. 

2.  Reduce    |    to   its   lowest   terms.     §   to   its   lowest 
terms.     |  to  its  lowest  terms. 

3.  Reduce  to  their  lowest  terms,  |,  •§ . 

4.  Reduce  to  their  lowest  terms,  j2^,  j5^. 

5.  Reduce  to  their  lowest  terms,  y4^,  T%  -j%,  -£,. 

6.  Reduce  to  their  lowest  terms,  y^,  y%,  y%,  -f%,  T92,  yf  • 

7.  Reduce  to  their  lowest  terms.  ^,  -f~,  y%,  y^,  |f ,  yf . 


94  RAY'S    INTELLECTUAL   ARITHMETIC. 

8.  Reduce  to  their  lowest  terms,  T%,  T66,  T%,  ±g,  ft,  ft. 

9.  Reduce  to  their  lowest  terms,  T2g,  T4g,  7*g,  T8g,  ig,  if. 

10.  Reduce  to  their  lowest  terms,  365,  o3T,  ^,  £g,  i£,  if. 

11.  Reduce  to  their  lowest  terms,  ||,  if,  if,  § j,  f {j,  f |, 


SECTION  XVII. 

LESSON  I. 

1.  David  divided  an  orange,  giving  to  William  1 -fifth, 
and  to  Sarah  2-fifths :  how  many  fifths  did  he  give  away  ? 
how  many  fifths  had  he  left  ? 

2.  After  taking  from  any  thing  3-fifths  of  itself,  what 
part  will  be  left? 

3.  John  bought  a  quart  of  chestnuts  :  he  gave  2-sixths 
to  Mary,  and  3-sixths  to  Eliza :  how  many  sixths  did  he 
give  away  ?     what  part  had  he  left  ? 

4.  One-fifth,  2-fifths,  and  7 -fifths   of  an   orange,  are 
how  many  fifths  ?     how  many  oranges  ? 

5.  How  much  are  i,  f,  and  J? 

6.  One -eighth,  3-eighths,    and   7-eighths  of  a  dollar 
are  how  many  eighths?     how  many  dollars? 

7.  What  is  the  sum  of  f ,  |,  and  |  ? 

8.  Daniel's  mother  gave  him  $J,  and  his   uncle  $J: 
how  many  sixths  of  a  dollar  did  he  receive  from  both  ? 

ANALYSIS. —  One-half  is  3-sixths,  and  \-third  is  2-sixths : 
3-sixths  and  2-sixths  are  5-sixths.  Ans.  5-sixths  of  a  dollar. 

The  fractions  are  reduced  to  a  common  denominator  before 
adding,  because  you  can  not  add  things  of  different  kinds. 
This  is  the  first  step  in  adding  or  subtracting  fractions. 

9.  Lucy  divided  an  orange,  giving  to  her  sister  1 -third, 
and  to   her   brother   1 -sixth  :    how  many   sixths  did  she 
give  ?     how  many  sixths  did  she  have  left  ? 


FRACTIONS.  95 

10.  James  bought  a  lemon,  and  gave  to  Lucy  1 -eighth, 
and  to  Susan  1 -fourth:  how  many  eighths  did  he  give? 
what  part  did  he  retain  ? 

11.  William    cut    a    pine-apple,    and    gave    to    Mary 
1 -third,  and  to  Eliza  1-ninth:  how  many  ninths  did  he 
give  to  both  ?     what  part  did  he  retain  ? 

12.  Thomas  bought  a  copy-book  for  fty1^,  and  a  reader 
for  $J :  how  many  tenths  of  a  dollar  did  they  both  cost? 

13.  David  gave  1 -fourth  of  a  melon  to  Eliza,  1 -third 
to  his  mother,  and  kept  the  remainder  :  how  many  twelfths 
did  he  give  ?     what  part  did  he  retain  ? 

14.  I  bought  1^-  yards  at  one  store,  and  2J  yards  at 
another :  how  many  yards  did  I  purchase  ? 

15.  I  planted  2J  acres  of  ground  in  corn ;  8|  acres  in 
oats  :  how  many  acres  were  in  both  pieces  ? 

16.  John  bought  a  knife  for  $J,  a  slate  for  $J,  and  a 
book  for  $f :  how  much  did  the  whole  cost  ? 

17.  Add    A  and    f.  £  and  f .         f  and  |. 

18.  Add    -|  and    f .  ±  and  i.         \  and  \. 

19.  Add    i  and    4.  f  and  If.         |  and  f . 

20.  Add    |  and    f .  f  and  f .         f  and  f . 

21.  Add  1|  and  2i.  1,     1,  |,  and     f 

22.  Add  3f  and  4|.  f ,     |,  |,  and     £. 

23.  Add  4f  and  5f  1',  2|,  |,  and6T%. 

24.  Add  5|  and  4|.  |,  3j,  4,  and  5  f . 

LESSON  II. 

1.  Mary  had  3-fourths  of  an  orange ;  she  gave  her 
uister  1 -fourth  :  how  many  fourths  had  she  left? 

2.  If  you  take  1-fifth  from  3-fifths,  what  will  be  left  ? 

3.  I  bought  5-sixths   of   a  quart  of  nuts,  and  gave 
3-sixths  to  my  sister  :  how  many  sixths  had  I  left  ? 

4.  One  is  how  many  sixths  ?     If  you  take  5-sixths 
from  1,  what  will  be  left? 


96  RAY'S   INTELLECTUAL   ARITHMETIC. 

5.  Two  is   how  many   fifths?     If   you   take   3-fifths 
from  2,  what  will  be  left  ? 

6.  If  you  take  i  from  f  ,  what  will  be  left  ? 

7.  |  from  f?    '  10.    T%  from  Jfc? 

8.  |  from  I?  11.    -|   from    1? 

9.  §  from  I  ?  12.     f   from   2  ? 

13.  Thomas   divided   an   orange,    giving    his    brother 
1-half,  and  his  sister  1  -third  :  how  many  sixths  did  each 
receive?     how  much  more  the  brother  than  the  sister? 

14.  If  a  bushel  of  wheat  cost  $J,  and  of  corn  $J,  how 
much  will  the  wheat  cost  more  than  the  corn  ? 

15.  If  you  take  \  from  f,  what  will  remain  ? 

16.  Joseph  bought  a  quart  of  chestnuts,  and  gave  1-half 
of  them  to  his  mother,  and  1-sixth  to  his   sister  :  how 
many  sixths  did  he  give  his  mother  more  than  his  sister  ? 

17.  Take  \  and  \  from  1,  what  will  be  left? 

18.  Jane    divided    an    orange,    giving    to    her    sister 
3-eighths,  and  to  her  brother  1  -fourth  :  to  which  did  she 
give  the  most  ?     what  part  of  the  orange  was  left  ? 

19.  A  man  having  72  miles  to  travel,  went  1  -third  the 
distance  the  first  day,  2-ninths  the  second,  the  remainder 
the  third  day  :    how  many  ninths  of  the  distance  did  he 
travel  further  on  the  first  day,  than  on  the  second  ?    what 
part  did  he  travel  the  last  day  ? 

20.  If  you  take  f  from  \,  what  will  be  left?    If  you 
take  |  and  -J  from  1,  what  will  be  left? 

21.  If  you  take  1  from  ^,  what  will  be  left? 

ANALYSIS.  —  \-fiftli   is    2-tenths,    and    l-half   is   5-tenths; 
2-tenths  from  5-tenths  leave  3-tenths.     Ans.  3-tenths. 


What  will  be  left  if  you  take 
om  J  ?         I  from  |  ?         1  fr 
om  £  ?         f  from  £  ?         £  fr 
24.     *  from  |  ?         4  from  f  ?         I  from  f  ? 

o  4  o  55  O 


22.  |  from  J  ?         I  from  |  ?         1  from  \  ? 

23.  1  from      ?  from      ?  from  f  ? 


FRACTIONS.  97 

25.  I  from  f  ?         J>  from  f  ?         ^  from    -|  ? 

26.  §  from  f  ?         f  from  |  ?         f  from     |  ? 

27.  J  from  f  ?         |  from  f  ?         *  from  2£  ? 

LESSON  III. 

1.  Mary  divided  a  quart  of  pecans,  giving  Ann  1 -third, 
and  Jane  1-fourth  of  them  :  what  part  had  she  left? 

2.  A  farmer  had  1^  bushels  of  wheat :  he  gave  to  1 
poor  man  J  of  a  bushel,  and  to  another  |  of  a  bushel : 
how  much  wheat  was  left  ? 

3.  James  had  ^   of  a  pound  of  raisins  :  he  gave  to 
his  brother  i  of  a  pound,  and  to  his  sister  \  of  a  pound : 
how  much  had  he  left  ? 

4.  If  from  3^  bushels  of  corn,  1^  bushels  be  taken,  how 
much  will  there  be  left? 

5.  A  lady   bought  3|   yards   of  muslin  at  one  store, 
and  2\  yards  at  another :  after  using  14  yards,  how  much 
had  she  left  ? 

6.  William's  father  gave  him  $| :  he  gave  to  a  poor 
person  $|,    for  apples  S-Jg,  and  for  a  book  $| :  what  part 
of  a  dollar  had  he  left  ? 

7.  James's  mother  gave  him  a  book :  he  read  the  first 
day|,  ^he  second  | ,   the   third  ^,  and  the  fourth  the  re- 
mainder :  what  part  did  he  read  the  fourth  day  ? 

8.  A  farmer  has  a  flock  of  84  sheep  in  4  fields :  the 
first  contains  i,  the  second  J,    and   the  third  \  of  them : 
what  part  does  the  fourth  field  contain  ? 

9.  Daniel   spends   -A    of  his  time  in  sleep,  \  of  it  at 
school,    y1^  in  reading,  and  ^    in   learning  music:  what 
part  of  his  time  is  not  employed? 

10.  A  pole  is  standing  in  a  pond;  ^  of  it  is  in  the  air, 
pud  |  in  the  water  :  what  part  is  in  the  earth  ? 

11.  A   student   devotes  |   of   his  time  to  sleep,  -|%  to 
Btudy,  75^  to  reading,  ^  to  exercise,  and   TV   to    deeds  of 
charity  7  what  part  of  his  time  is  unemployed  ? 


98  RAY'S   INTELLECTUAL   ARITHMETIC. 

12.  One-third  of  an  orchard  is  apple  trees,  ^  pear  trees, 
|  plum  trees,  7V  quince  trees,  and  the  remainder  peach 
trees :    if  the   orchard  contain   96  trees,   how  many  are 
there  of  each  kind  ? 

13.  After   spending   ^   and  -J-  of  my  money,  and  los- 
ing jV,  I  had  §8  remaining :  how  much  had  I  at  first? 

14  .The    difference    between    |  and   i   of  my  money 
is  $3  :  how  much  have  I  ? 

15  .1  ate  f  of  my  peaches,  gave  away  10,  and  have  10 
left :  how  many  had  I  at  first  ? 


SECTION  XVIII. 

LESSON  I. 

1.  A  father  gave  each  of  his  two  sons  half  a  dollar: 
how  much  did  he  give  them  both  ? 

2.  A   mother   gave    each  of  her   3  children   half  an 
orange :  how  many  half  oranges  did  it  take  ?     how  many 
oranges  ?     why  ? 

3.  What  are  3  times  1-half?     4  times  1-half? 

4.  John  fed  5  horses,  giving  to  each  half  a  peck  of 
oats  :  how  many  half  pecks  did  it  take  ?   how  many  pecks  ? 
why? 

5.  What  are  5  times  1-half?     6  times  1-half? 

6.  What  are  8  times  1-half?     9  times  1-half? 

7.  James  gave   1 -third  of  an  orange  to  each  of  his 
sisters  :  how  much  did  he  give  to  both  ?     why  ? 

8.  If  a  man  can  eat  1 -third  of  a  pound  of  meat  in  1 
day,  how  much  can  he  eat  in  3  days  ?     why  ? 

9.  What  are  4  times  1 -third  ?     5-times  1 -third  ? 

10.  John  gave  2-thirds  of  a  pine-apple  to  each  of  his  2 
brothers  :  how  many  thirds  did  he  give  to  both  ?  How 
many  pine-apples  did  it  take  ?  Ans.  1  and  1-third. 


FRACTIONS. 


99 


11.  What  are  4  times   2-thirds?      5  times   2-thirds? 

12.  Thomas  gave  1 -fourth  of  an  apple  to  each  of  his  3 
playmates  :  how  many  fourths  did  it  take  ?     why  ? 

13.  If  1  bushel  of  oats  cost  1 -fourth  of  a  dollar,  how 
much  will  4  bushels  cost  ?     why  ? 

14.  Charles  gave   3-fourths   of  a   pint  of  chestnuts  to 
each  of  his  2  brothers  :  how  many  fourths  of  a  pint  did 
it  take  ?     how  many  pints  ?     why  ? 

15.  Mary  gave  3-fourths  of  an  orange  to  each  of  her 
three  brothers :   how   many  fourths   of  an   orange  did  it 
take  ?     how  many  oranges  ?     why  ? 

16.  What  are  5  times  3-fourths  ? 

ANALYSIS. — 5  times  3-fourths  are  I5-fourths ;  15-fourths 
are  as  many  ones  as  4-fourths  are  contained  times  in 
15-fourths ;  4-fourths  in  15-fourths,  3  and  3-fourths  times. 
Ans.  3  and  3-fourths. 

17.  What  are  6  times  |  ? 

18.  What  are  3  times   |  ? 

19.  What  are  3  times  4  ? 

20.  What  are  5  times  £  ? 

21.  What  are  2  times  ^  ? 
What  are  8  times  f  ? 
What  are  5  times   X  ? 


22. 
23. 
24. 
25. 

26. 


What  are  7  times  f  ? 
What  are  3  times 
What  are  7  times 


3? 
I? 


7 

times 

| 

? 

8 

times 

<i 

? 

5 

times 

2 

? 

6 

times 

5 

? 

4 

times 

§ 

? 

5 

times 

4 

? 

2 

times 

| 

? 

4 

times 

F 

? 

4 

times 

~y 

? 

5 

times 

| 

V 

6 

times 

5 

? 

7 

times 

§ 

? 

8 

times 

! 

9 

9 

times 

3 

? 

6 

times 

7 

? 

8 

times 

-|- 

? 

4 

times 

f 

? 

6 

times 

8 
7 

? 

8 

times 

4 

? 

9 

times 

| 

? 

4 

times 

fr 

-? 

7 

times 

rt 

.! 

27.   What  are  3  times  T30?    4  times 

LESSON  II. 

1.  William   gave   an   orange   and   a  half  to  each   of 
his  2  sisters  :  how  many  oranges  did  it  take  ? 

2.  What  are  2  times  1  and  1-half  ? 

3.  What  are  3  times  H  ?    4  times  11  ?   5  times  l-.J  ? 


100  BAY'S   INTELLECTUAL   ARITHMETIC. 

AXAL. — 3  times  I  are  3,  and  3  times  l-half  are  3-halves, 
equal  1  and  l-half;  this  added  to  3  makes  4  and  l-half. 

4.  If  1  bushel  of  wheat  cost  $1  and  1 -third  of  a  dol- 
lar, what  will  2  bushels  cost  ? 

5.  How  many  are  3  times  lJ  ?  2  times  2f  ? 

6.  How  many  are  3  times  3|  ?  4  times  4^  ? 

7.  How  many  are  5  times  2|  ?  6  times  3|  ? 

8.  How  many  are  8  times  3|  ?  9  times  4|  ? 

9.  If  1  bushel  of  barley  cost  $1   and  1-fourth   of  a 
dollar,  what  will  3  bushels  cost  ?     4  bushels  ? 

10.  How  many  are    5  times  1|  ?  6  times  1|  ? 

11.  How  many  are    2  times  1|  ?  3  times  2|  ? 

12.  How  many  are    4  times  3i  ?  5  times  3|  ? 

13.  How  many  are    6  times  3|  ?  8  times  3|  ? 

14.  How  many  are    7  times  2|  ?  9  times  2|  ? 

15.  How  many  are  10  times  1|  ?  10  times  3^  ? 

16.  How  many  are  11  times  2|  ?  12  times  3|  ? 

17.  If  a  family  consume  3  and  1 -fifth  barrels  of  flour 
io  one  month,  how  much  will  they  require  for  3  months  ? 

18.  How  many  are  4  times  3|  ?  5  times  3|  ? 

19.  How  many  are  2  times  6f  ?  3  times  2|  ? 

20.  How  many  are  6  times  4i  ?  6  times  3|  ? 

21.  How  many  are  7  times  4|  ?  8  times  3|  ? 

22.  How  many  are  9  times  If  ?  9  times  3 \  ? 


SECTION  XIX. 


LESSON  I. 

1.  If  you  divide  an  apple  into  two  equal  parts,  >yhat 
I  part  called?     what  is  l-half  of  1  ? 


FRACTIONS.  101 

2.  If  you  divide  3  apples  between  2  boys,  how  many 
apples  will  each  have  ?  how  will  you  divide  them  ?  what 
is  1-half  of  3? 

ANALYSIS.  —  l-half  of  I  is  l-half,  and  l-half  of  L  is  3 
times  l-half  of  1,  which  are  3-halves  ;  3-halves  are  as  many 
ones  as  2-halves  are  contained  times  in  %-halves;  2-halves  in 
Sj  1  and  l-half  times.  Ans.  1  and  l-half 


3.  John  bought  5  oranges,  and  divided  them  between 
his  2  sisters  :  how  many  oranges  did  each  have  ? 

4.  What  is  l-half  of  6?    f  of  7?    Jof8?    Jof9? 
of  11?     of  13? 

5.  If  cloth  is  SI  a  yard,  what  must  be  paid  for  1  -third 
of  a  yard  ?     what  is  1  -third  of  1  ? 

6.  James  has  2  oranges  to  divide  among  his  3  sisters  : 
how  must  he  divide  them,  and  what  part  of   an  orange 
will  each  have?     what  is  1  -third  of  2? 

7.  If  3  bushels  of  pears  cost  $4,  how  much  is  that  a 
bushel  ?     what  is  1  -third  of  4? 

ANALYSIS.  —  l-third  of  4  is  the  same  as  4  times  l-third  of 
1,  which  are  4-thirds  ;  4-thirds  are  as  many  ones  as  3-thirds 
are  contained  times  in  4-thirds:  3-thirds  w  4-thirds  1  and 
l-third  times.  Ans.  1  and  l-lhird. 

8.  A  carpenter  receives  ?5   for  3  days'  work:   how 
much.  is  that  a  day?     what  is  l-third  of  5? 

9.  What  is  l-third  of  6?    J  of  7  ?    J  of  8  ?     Jof9? 
of  10?     of  11?     of  13? 

10.  What  is  l-third  of  16?     of  17?     of  18?    of  19? 
of  20  ?    of  21  ?     of  22  ? 

11.  What  is  l-third  of  23?     of  24?     of  25  ?     of  26? 
of  27?    of  28?     of  31? 

12.  If  4  bushels  of  oats  cost  $1,  what  part  of  a  doL 
lar  will  1  bushel  cost?    what  is  1  -fourth  of  1  ? 

13.  A  mother  divided  3  pears   equally  among  her  4 
children,  what  part  of  a  pear  did  each  receive  ? 

2d  Bk.  7 


102  BAY;S   INTELLECTUAL   ARITHMETIC. 

14.  What  isl-fourthof  4?    Jof5?    Jof6?    of  7? 
of  8?    of  9?     of  10?    of  11? 

15.  What  is  1 -fourth  of  12  ?    of  13  ?    of  15  ?    of  17  ? 
of  19?    of  23? 

16.  What  is  1 -fourth  of  25  ?    of  26  ?     of  27  ?     of  28  ? 
of  29  ?     of  30  ?     of  31  ? 

17.  If  a  melon  cost  5  cents,  what  part  of  a  melon  can 
you  buy  for  1  cent?     what  is  1 -fifth  of  1  ? 

18.  James  had  2  pine-apples,  and  divided  them  among 
5  of  his  companions :  what  part  of  a  pine-apple  did  each 
have  ?    what  is  1 -fifth  of  2  ? 

19.  What  is  1-fifth  of  3?     £of4?     i  of  5  ?     of  6? 
of  7?    of  8.?    of  9?    of  10?    of  11?    of  12? 

20.  What  is  1-sixth  of  1  ?    J  of  2  ?     I  of  3  ?     of  4  ? 
of  5?    of  6?    of  7?    of  8?    of  10?    of  15? 

21.  What  is  1-ninth  of  2?     '  of  4?     ^of7?     of  8? 
of  9?    of  12?    of  15?    of  17?    of  19? 

22.  How  do   you   find  one-half  of  any  thing?     one- 
third?     one-fourth?     one-fifth?     one-eighth? 

LESSEN  II. 

1.  If  3  bushels  of  wheat   cost  $1,  what  part  of  $1 
will  1  bushel  cost  ?    what  part  will  2  bushels  cost  ? 

2.  What  is  1 -third  of  1  ?     what  are  2-thirds  of  1  ? 

3.  If  3  bushels  of  wheat  cost  $2,  what  part  of  81 
will  1  bushel  cost  ?     what  part  will  2  bushels  cost  ? 

ANAL. — 1  bushel  will  cost  l-third  of  $2,  which  is  2-thirds; 
and  2  bushels  will  cost  twice  as  much  as  1  bushel;  that  is> 
2  times  2-thirds,  which  are  4-thirds  of  $1,  equal  $1  and 
l-third  of  a  dollar.  Ans.  $1  and  l-third. 

4.  If  3  barrels  of  cider  cost  $4,  what  part  of  a  dol- 
lar will  1  barrel  cost  ?     what  part  will  2  barrels  cost  ? 

5.  What  is  l-third  of    4  ?  2-thirds  of    4  ? 

6.  What  is  l-third  of    5  ?  2-thirds  of    5  ? 


FRACTIONS.  103 

7.  What  is  1-third  of    6?  2-thirds  of    6? 

8.  What  is  1-third  of    7  ?  2-thirds  of    7  ? 

9.  What  is  1-third  of    8  ?  2-thirds  of    8? 

10.  What  is  1-third  of    9  ?  2-thirds  of    9  ? 

11.  What  is  1-third  of  10  ?  '  2-thirds  of  10  ? 

12.  If  4  barrels  of  apples  cost  $3,  what  part  of  a  dol- 
lar will  1  barrel  cost  ?     2  barrels  ?     3  barrels  ? 

,    13.  If  5  apples  cost  2  cents,  what  part  of  a  cent  will 
1  apple  cost  ?    2  apples  ?    3  apples  ?     4  apples  ? 

14.  What  are  2-fifths        of  13  ?  3-fifths        of  13  k 

15.  What  is    1-sixth        of    7  ?  5-sixths      of    7  ? 

16.  What  are  3-sixths       of  10  ?  4-sixths      of  10? 

17.  What  is    1-seventh    of    9  ?  3-sevenths  of    9  ? 

18.  What  are  2-sevenths  of  11  ?  3-sevenths  of  12  ? 

19.  What  is    1-eighth      of  13  ?  3-eighths    of  13? 

20.  What  are  5-thirds       of    4  ?  7-thirds      of    5  ? 

21.  What  are  7-eighths    of  10  ?  9-eighths    of    3  ? 

22.  What  are  3-tenths     of    7  ?  7-tenths      of    3  ? 

23.  What  are  5-ninths      of  11  ?  9-fifths        of    7  ? 

24.  Which  is  the  greater  4  of  2.  or  *  of  1  ?    i  of  3. 
or  f  of  1  ? 

LESSON   III. 

1.  James  bought  3  lemons  for  7  cents:    how  much 
was  that  apiece? 

2.  William  bought  5  quarts  of  chestnuts  for  18  cents : 
at  that  rate,  what  was  the  cost  of  2  quarts  ? 

3.  If  7  pounds  of  cheese  sell  for  40  cents,  how  much 
should  5  pounds  sell  for? 

4.  Bought  6  yards  of  muslin  ;  gave  my  mother  3-fifths, 
and  kept  the  remainder  :  how  many  yards  had  each  ? 

5.  If  a   quantity  of  provisions  serve   2  men  7  days, 
how  long  will  it  last  1  man  ?    how  long  3  men  ? 


104  RAY'S  INTELLECTUAL  ARITHMETIC. 

6.  If  4  men  perform  a  job  of  work  in  8  days,  how 
long  will  it  require  5  men  ? 

ANALYSIS. — It  will  take  1  man  4  times  as  long  as  4  men  ; 
*,nd  5  men  \-fifth  as  long  as  1  man. 

7.  If  a  barrel  of  cider  will  last  5  men  8  days,  how 
long  will  it  last  3  men  ? 

8.  If  1  barrel  of  flour  serve  8  persons  10  days,  how 
long  will  it  last  11  persons? 

9.  If  7  men  can  do  a  piece  of  work  in  5  days,  how 
long  will  it  require  8  men  ? 

10.  If  2  men  build  a  wall  in  12  days,  how  long  will 
it  take  7  men  ? 

11.  If  it  requires   11  days,  of  8  hours  each,  to  do  a 
job,  how  many  will  be  required  of  10  hours  each? 

12.  A  man  paid   37  cents  for   riding  8  miles :  at  the 
same  rate,  what  will  it  cost  to  ride  11  miles? 

13.  Two  pipes  of   a  certain  size  will  empty  a  cistern 
in  17  minutes :  in  what  time  will  3  pipes  empty  it  ? 

14.  If  18  bushels  of  oats  last  5  horses  1  week,  how 
many  bushels  will  7  horses  require  ? 

15.  If  a  laborer  receive  5  bushels  of  wheat  for  7  days' 
work,  how  much  should  he  receive  for  11  days? 

16.  If  a  carpenter  earn  $8  in  5  days,  how  much  will 
he  earn  in  9  days  ? 

17.  If  3  yards  of  cloth  cost  $13,  what  cost  8  yards? 

18.  A  pole,   18  feet  long,  is   2-sevenths  in  the  earth, 
the  rest  in  the  air :  what  is  the  length  of  each  part  ? 

19.  Three  men,  A,  B  and  C,  found  a  bag  containing 
815:    A   got  2-ninths,  B  1-third,  and  C  the  remainder: 
what  was  the  share  of  each  ? 

20.  If  J  a  yard  of  cloth  cost  §4,  what  cost  6  yards  ? 

21.  If  3  ounces  cost  36  cents,  what  should  be  charged 
for  §  of  an  ounce  ? 

22.  If  |  of  a  number  equal  18,  what  is  the  |  of  it? 


FRACTIONS.  ,05 

23.  How  much  must  a  man  earn  a  day  to  receive  $72 
for  8  weeks,  6  days  to  a  week  ? 


SECTION   XX. 

LESSON  I. 

ILLUSTRATION. — If  1-half  of  a  yard  of  tape  be  divided 
into  2  equal  parts,  one  of  the  parts  is  1-half  of  1-half; 
that  is,  1 -fourth  of  the  whole  yard. 

one-half.  one-half. 


one-fourth.      one-fourth.      one-fourth.      one-fourth. 

If  1 -third  of  a  yard  be  divided  into  two  equal  parts, 
one  of  the  parts  is  1 -sixth  of  the  whole  yard;  that  is, 
1-half  of  1-third,  is  1-sixth. 

one-third.  one-third.  one-third. 


l-sixth.     1-sixth.     1-sixth.     1-sixth.     1-sixth.     1-sixth. 

1.  Mary  having  1-half  of  an  orange,  gave  her  brother 
1  -half  of  what  she  had :  what  part  of  a  whole  orange  did 
she  give  him  ? 

2.  James  divided  1-third  of  an  apple  equally  between 
his  2  brothers  ;  what  part  did  each  receive  ? 

3.  What  is  J  of  |?       i  of  £? 

ANALYSIS. — 1-half  of  1-third  is  the  fraction  obtained  by 
dividing  a  unit  into  3  equal  parts,  and  one  of  these  parts 
into  2  equal  parts ;  but  if  a  unit  (one)  be  thus  divided,  there 
will  be  6  parts,  and  each  part  will  be  1-sixth  of  the  whole; 
therefore,  1-half  of  1-third  is  1-sixth.  < 


106  RAY'S   INTELLECTUAL   ARITHMETIC. 

4.  Thomas  divided  1-half  of  a  lemon  equally  between 
his  3  sisters :  what  part  did  each  receive  ? 

5.  If  1 -fourth  of  an  orange  be  divided  into  2  equal 
parts,  what  is  1  of  the  parts  called  ? 

6.  What  is  J  of  J  ?       J  of  J  ? 

7.  If  1- third  of  an  apple  be  cut  into  3  equal  parts, 
what  part  of  the  apple  will  each  piece  be  ? 

8.  If  each  half  of  an  apple  be  divided  into  5  equal 
parts,  how  many  parts   will   there   be  ?     what  is  1   part 
called  ?     what  is  4  of  £  ? 

9.  If  you  divide  an  orange  into  4  equal  parts,  and  cut 
each  part  into  3  pieces,  what  is  1  piece  called  ? 

What    single    Fraction    equals 


10.  I  of 

11.  i  of 

12.  4  of 


of  |? 
on? 
?        4  of  4?        4  of  4?        4  of  A? 


4     «*•     5  '  3    «A    g  '  5    U1     5   '  3    UI     9 

is.  ion?    i°fi?    i°u?    i°f4? 

14.  Thomas  has    2-thirds  of  an   apple,  and  wishes  t<? 
give  his  brother  1-half  of  what  he  has:  what  part  of  the 
whole  apple  must  he  give  him  ? 

15.  What  is  J  of  J  ?       J  of  §  ? 

ANALYSIS. — I-half  of  2-thirds  is  2  times  l-half  of  1-third; 
\-half  of  l-third  is  l-sixth;  2  times  l-sixth  are  2-sixths. 

16.  Daniel  has  3-fifths  of  a  melon  to  divide   equally 
between  his  brother  and  sister :    how  must  he  divide  it 
and  what  part  of  the  whole  will  each  receive  ? 

17.  What  is  i  of  |  ?  i  of  |  ?  i  of  |  ? 

18.  What  is  |  of  |  ?  1  of  §  ?  i  of  |  ? 

19.  What  is  I  of  4  ?  4  of  g  ?  -i  of  «  ? 

20.  What  is  4  of  |  ?  £  of  3  ?  £  of  f  ? 

21.  Edward  has  4-fifths  of  a  melon,  and  wishes  to  give 
his  sister  2-thirds  of  what  he  has  :  what  part  of  the  whole 
melon  will  she  receive  ? 


FRACTIONS.  107 

22.  What'is  £  of  T%  ?      |  of  f  ?      f  of  f  ? 

ANALYSIS. — 2-thirds  of  4-jfifths  are  2  times  1 -third  of 
trfifths,  and  l-third  of  4-ffths  is  4  times  l-third  of  l-Jifth. 

23.  What  are  f  of  f  ?  |  of   f  ?  |  of   f  ? 

24.  What  are  |  of  jj  ?  |  of  f  ?  |  of  |  ? 

25.  What  are  f  of  f  ?  |  of  4  ?  |  of  Ty 

26.  What  are  f  of  f  ?  f  of  _5_?  f  of  f  ? 

27.  If  1  yard  of  cloth  is  worth  2J  bushels  of  wheat, 
what  is  l-half  of  a  yard  worth  ? 

ANALYSIS. — 2  and  l-half  are  5-halves,  and  l-half  of  5-halves 
is  5  times  l-half  of  l-half ,  or  5-fourths,  equal  1  and  l-fourth. 
Ans.  1  and  l-fourth  bushels. 

What  single  Fraction  will  represent 

28.  i  of  2-i  ?      i  of  11  ?       J  of  If  ?       |  of  2£? 

29.  A  of  2|  ?      i  of  3i  ?       i  of  4|  ?       i  of  51  ? 

30.  f  of  U  ?      |  of  1|  ?      f  of  LI  ?       |  of  1|  ? 

31.  |  of  21?       |  of  4|  ?      |  of  2|?       I  of  3£? 

LESSON  II. 

1.  A  person  owning  |  of  a  ship,  sold  |  of  his  share : 
what  part  of  the  whole  ship  did  he  sell  ? 

2.  If  3  yards  of  cloth  cost  $2|,  what  cost  2  yards? 

ANALYSIS. — 1  yard  will  cost  l-third  as  much  as  3  yards, 
and  2  yards  will  cost  2  times  as  much  as  1  yard;  that  is, 
2-thirds  as  much  as  3  yards. 

3.  If  2  yards  of  cloth  cost  $1|,  what  cost  3  yards? 

4.  If  3  yards   of  cloth  cost  $5^,  what  part  of  that 
sum  will  2  yards  cost? 

5.  If  5   gallons  of  molasses  cost  $3§,  what  part  of 
that  sum  will  3  gallons  cost  ? 


108  KAY'S  INTELLECTUAL  ARITHMETIC. 

6.  If  7  pounds  of  sugar  cost  §l-i,  what  cost  4  pounds  ? 

7.  If  8  pounds  of  butter  cost  $1J ,  what  cost  7  pounds  ? 

8.  If  7  yards  of  cloth  cost  $5|,  what  will  be  the  cost 
of  3  yards  ?     of  4  yards  ? 

9.  If  3   barrels   of  cider   cost  $4|,  what  part  of  that 
sum  will  5  barrels  cost? 

10.  If  5  gallons  of  oil  cost  $2|,  what  part  of  that  sum 
will  7  gallons  cost  ? 

11.  If  2  gallons  of  molasses  cost  $1|,  what  will  be  the 
cost  of  3  gallons  ? 

12.  If  3  bottles  of  wine  cost  $2|,  what  will  be  the  cost 
of  8  bottles?     of  10  bottles? 

13.  If  2  men  do  a  job  of  work  in  3|  days,  how  many 
days  will  it  take  1  man  ?     3  men  ? 

14.  A  man  can  do  a  job  in  3|  days,  of  10  hours  each: 
how  many  days  will  it  take  of  7  hours  each  ? 

15.  If  a   man  can  do  a  piece  of  work  in  15|  days, 
working  5   hours   a  day,   how  many   days  will    it   take, 
working  8  hours  a  day  ? 


SECTION  XXI. 

LESSON   I. 

1.  A  boy  had  2-thirds  of  an  orange,  which  he  divided 
equally  between  his  two  sisters :  what  part  of  an  orange 
did  each  receive  ? 

ANALYSIS. — Since  2  apples  divided  by  2  give  1  apple,  and 
$2  divided  by  2  give  $1,  it  follows  that  2-thirds  divided  by  2 
must  give  l-third.  Each,  therefore,  must  have  received  l-third 
of  an  orange. 

2.  How  often  is  3  contained  in  3-fourths  ? 

3.  If  2  and  2-thirds  dollars  be  divided  among  4  men, 
what  is  each  man's  share  ? 


FRACTIONS.  109 

4.  If  2f  be  divided  by  7,  what  will  be  the  result  ? 

5.  How  many  times  6,  in  3  and  3-fifths  ? 

6.  How  many  thirds  of  9,  are  contained  in  3|  ? 

7.  Divide  6  and  3-fourths  cents  among  9  boys. 

8.  A  man  having  lOf  acres  of  land,  divided  it  equally 
between  his  5  children :  how  much  did  each  receive  ? 

9.  If  4f  be  divided  by  11,  what  will  be  the  quotient? 

10.  If  $|  be  equally  divided  among  3  men,  what  part  of 
a   dollar   will  each  get  ? 

11.  How  divide  a  fraction  by  any  whole  number  ? 

LESSON  II. 

1.  If  1  apple  cost  i  a  cent,  how  many  will  1  cent  buy? 
why  ?      Ans.    Because  there  are  2  half -cents  in  1  cent. 

2.  If  1  pear  cost  1- third  of  a  cent,  how  many  pears 
can  you  buy  for  1  cent?     why? 

3.  When  1  bushel  of  wheat  costs  &|,  how  many  bushels 
can  you  buy  for  $H  ? 

4.  How  often  is  ^  contained  in  2i  or  f  ? 

ANALYSIS. —  One-half  is  contained  in  Fiw-halves,  as  often 
as  1  is  contained  times  in  5  ;  that  is,  5  times :  for,  if  1  apple 
is  contained  in  5  apples  5  times ;  1  cent  in  5  cents,  5  times; 
\-half  must  be  contained  in  5-halves,  5  times. 

5.  If  1  lemon  cost  half  a  cent,  how  many  lemons  can 
be  bought  for  5  cents  ? 

6.  If  1  apple  cost  1 -third  of  a  cent,  how  many  apples 
can  John  buy  with  4  and  1 -third  cents? 

7.  If    1    peach   cost   2-thirds    of   a    cent,  how  many 
peaches  can  you  purchase  for  4  and  1 -third  cents? 

ANALYSIS. — 4  and  l-third  are  13-thirds  ;  and  Two-thirds  are 
contained  in  THiRTEEN-rffaVds  as  often  as  2  is  contained  in  13, 
that  is,  6  and  \-half  times.  Ans.  6^  peaches- 


HO  RAY'S   INTELLECTUAL   ARITHMETIC. 

8.  I  distributed  2f  bushels  of  wheat  among  a  number 
of  poor  persons,  giving   to   each,    2-thirds  of  a  bushel  : 
now  many  persons  were  there  ? 

9.  At  §|  a  yard,  how  many  yards  of  calico  can  be 
purchased  for  &3f  ? 

10.  At   $|  a  yard,  how   many  yards  of  ,  cloth  can  be 
purchased  for  S3  ]  ? 

11.  If  a  lemon   cost  3-fourths  of  a  cent,  how  many 
can  be  purchased  for  3|  cents  ?     for  5£  cents  ? 

12.  One  bushel  of  rye  is  worth  3-fourths  of  a  bushel 
of  wheat:    how   many    bushels    of   rye    can   be   bought 
with  4f  bushels  of  wheat  ?     with  8|  bushels  ? 

13.  At  1  -fifth  of  a  cent  each,  how  many  cherries  can 
Mary  purchase  with  3  cents  ?     why  ? 

14.  At  $?  a  gallon,  how  many  gallons  of  vinegar  can 
you  buy  for  ?2f  ?     for 


15.  How  often  is  f  contained  in  2|  ?     in  4f  ?     in  61  ? 

16.  How  often  is  J  contained  in  3|  ?     in  51  ?     in  4|  ? 

17.  How  often  is  f  contained  in  1  ?       in  3f  ?     in 

18.  How  often  is  f  contained  in  3  ?       in  4|  ?     in 

19.  At  $|  a  yard,  how  much  cloth  can  be  had  for 


ANALYSIS.  —  As  many  yards  as  l-third  of  a  dollar,  the  price 
of  1  yard,  is  contained  times  in  I  -half  of  a  dollar;  l-half  is 
equal  to  3-sixths,  and  l-third  to  2-sixths  :  2-sixth.s  in  3-sixths, 
1  and  l-half  times.  Ans.  1  and  l-half  yards. 

20.  At  $^  a    yard,  how  much  gingham  can   be    pur- 
chased   for    $|  ? 

21.  How  often  is  i  contained  in  |  ? 

22.  If  1  yard  of  calico  cost  $A,  how  much  can  be  pur- 
chased  for  $|  ? 

23.  How  often  is  i  contained  in  |? 

NOTE.  —  Reduce  the  fractions  to  a  common  denominator,  then 
divide  their  numerators. 


FRACTIONS.  1 1 1 

24.  How  often  is    ^  contained  in  |  ?  in  f  ?  in  {£  ? 

25.  How  often  is    -|  contained  in  f  ?  in  f  ?  in  }|  ? 

26.  How  often  is    f  contained  in  f  ?  in  |  ?  in  i|  ? 

27.  How  often  is    |  contained  in  f  ?  in  |  ?  in  T90  ? 

28.  How  often  is  U  contained  in  |  ?  in  |  ?  in  2|  ? 

29.  How  often  is  2\  contained  in  §  ?  in  |  ?.  in  3|  ? 

30.  How  often  is  3|  contained  in  |  ?  in  f  ?  in  5|  ? 


SECTION   XXII. 

LESSON  I. 

1.  James  gave  his  brother  1  apple  and  1-half,  which 
Was  1-half  of  what  he  had  :  how  many  had  he  ? 

2.  If  1-third  of  a  yard  cost  $l£,  what  cost  a  yard? 

3.  If  a  man  travel  1  and  1-third  miles  in  1-  ourth  of 
an  hour,  how  far  can  he  travel  in  1  hour  ? 

4.  Daniel  bought  1  -fifth  of  an  orange  for  3 J  cents  : 
at  that  rate,  what  will  a  whole  one  cost  ? 

5.  If  1 -fourth  of  a  yard  of  tape  cost  3  and  2-thirds 
cents,  what  will  1  yard  cost  ? 

ANALYSTS. — 3  and  2-thirds  are  l-fourth  of  4  times  3  and 
2-thirds  ;  4  times  2-thirds  are  S-thirds,  equal  2  and  2-thirds ; 
4  times  3  are  12,  which  added  to  2  and  2-thirds^  make  14 
und  2-thirds.  Ans.  14  and  2-thirds. 

6.  5  and  1-third        are  1-half       of  what  number? 

7.  7  and  3-fourths     are  1-third      of  what  number  ? 

8.  5  and  3-eighths    are  1 -fifth        of  what  number? 

9.  3  and  4-sevenths  are  1-seventh  of  what  number? 

10.  4  and  4-fifths        are  1-six.th       of  what  number? 

11.  9  and  2-thirds      are  1 -eighth    of  what  number? 


112  RAY'S    INTELLECTUAL    ARITHMETIC. 

12.  If  2-thirds  of  a  yard  of  ribbon  cost  3  cents,  what 
will  1-third  cost?     If  1 -third  of  a  yard  cost  1  and  1-half 
cents,  what  will  a  yard  cost  ? 

13.  One  and  1-half  is  1-third  of  what  number  ?     Three 
is  2-thirds  of  what  number  ? 

14.  If  2-thirds  of  a  barrel  of  flour  cost  $5,  what  cost 
1-third  of  a  barrel  ?     If  1-third  of  a  barrel  cost  $2J-,  what 
will  a  whole  barrel  cost  ? 

15.  Five  is  2  times  what  number  ? 

ANALYSIS. — 5  is  2  times  l-half  of  5;  1-half  of  5  is  2  and 
1-half;  therefore,  5  is  2  times  2  and  1-half 

16.  Five  is  2-thirds  of  what  number? 

ANALYSIS. — 5  is  2-thirds  of  that  number  of  which  1-halJ- 
of  5  is  1-third:  1-half  of  5  is  2  and  1-half  and  2  and  1-half 
are  1-third  of  3  times  2  and  1-half  which  are  7  and  1-half; 
therefore,  5  is  2-thirds  0/7  and  1-half 

17.  If  3-fourths  of  a  yard  of  velvet  cost  $4,  what  will 
1 -fourth  of  a  yard  cost?     If  1 -fourth  cost  $1|,  what  will 
a  whole  yard  cost  ? 

18.  Four  is  3-fourths  of  what  number  ? 

19.  If   2-fifths  of  a  quart  of   chestnuts  cost  3  cents, 
what  will  1 -fifth  cost?     If  1 -fifth  of  a  quart  cost  1  and 
1-half  cents,  what  will  a  whole  quart  cost? 

20.  Three  is  2-fifths  of  what  number  ? 

21.  If  4-fifths  of  a  yard  cost  $7,  what  will  1-fifth  cost? 
If  1-fifth  of  a  yard  cost  $1|,  what  cost  a  yard? 

22.  Seven  is  4-fifths  of  what  number  ? 

23.  If   3-eighths   of  a   melon  cost  4  cenfs,  what  will 
1-eighth  cost?     what  will  a  whole  melon  cost  ? 

24.  If  3-fifths  of  a  pound  of  sugar  cost  10  cents,  what 
will  1-fifth  cost  ?     what  will  1  pound  cost? 

25.  If  5-sixths  of  a  barrel  of  beef  cost  $6,  what  will 
1 -sixth  cost?     what  will  a  barrel  cost? 


FRACTIONS.  113 

26.  Fifteen       is  2-ninths  of  what  number  ? 

27.  Fourteen    is  3-eighths  of  what  number  ? 

28.  Thirteen     is  3-fourths  of  what  number  ? 

29.  Seventeen  is  8-ninths  of  what  number  ? 

30.  If  1  yard  and  a  half,  that  is,  3  half  yards  of  rib- 
bon cost  6  cents,  what  will  a  yard  cost  ? 

31.  Six  is  3-halves  of  what  number  ? 

ANALYSIS. —  Of  that  number  of  which  l-third  of  6  is 
l-half;  l-third  of  6  is  2 ;  2  is  \-lialf  of  2  times  2,  which  are 
4;  therefore,  6  is  3-halves  of  4. 

32.  If  1  yard  and  l-third  of  a  yard,  that  is,  4-thirds 
of  a  yard  of  cloth,  cost  $3,  what  will  a  yard  cost  ? 

33.  If  a  man  travel  9  miles  in  1  hour  and  2-sevenths 
of  an  hour,  how  far  will  he  travel  in  1  hour  ? 

34.  By  selling  a  watch  for  $18,  I  gained  1 -fourth  of 
what  it  cost  me  :  how  much  did  it  cost  ? 

35.  A  grocer,  by  selling  a  lot  of  flour  for  $25,  gained 
1 -fifth   of  what   it  cost  him:    what  was  the  cost?     how 
much  did  he  gain  ? 

36.  If  a  man  pay  $6  for  1  and  l-third  yards  of  cloth, 
what  is  the  cost  of  1  yard  ? 

37.  If  a   man   receive   $10  for  2  and   2-thirds   days' 
work,  how  much  is  that  per  day  ? 

38.  If  2-thirds  of  a  yard  cost  $4|,  what  cost  a  yard  ? 

39.  Four  and  6-sevenths,  are  2-thirds  of  what  ? 

ANALYSIS — Of  that  number  of  which  l-half  of  4  and  6-se& 
#nths  is  l-third. 

40.  Four     and  2-thirds    .are  2-fifths      of  what  ? 

41.  Three  and  3-fourths  are  5-sixths     of  what? 

42.  One      and  3-fifths      are  3-fifths       of  what? 

43.  Three  and  2-thirds     are  3-fourths  of  what  ? 

44.  Four    and  4-fifths      are  6-halves     of  what  ? 


114  RAY'S   INTELLECTUAL   ARITHMETIC. 


LESSON    II. 

1.  A  man  gave  to  some  poor  persons  $3,  which  was 
2-fifths  of  his  money :  how  much  had  he  left  ? 

2.  A  pole  stands  5 -sevenths  in  the  air,  and  5  feet  in 
the  ground :     how  long  is  the  pole  ? 

3.  A  man  spent  2-fifths  of  his  money,  and  had  $10 
left :     how  much  did  he  spend  ? 

4.  At  3  yards  for  2   cents,  how  many  yards  of  tape 
can  be  bought  for  7  cents  ? 

5.  At  5  lemons  for  3  cents,  how  many  lemons  can  be 
bought  for  12  cents  ? 

6.  If  4-fifths   of  a  yard  of  calico   cost  8  cents,  how 
many  yards  can  be  purchased  for  25  cents  ? 

7c  If  1  and  1 -fourth  tuns  of  hay  cost  $8,  what  is  the 
price  of  1  tun  ? 

8.  If  6-sevenths  of  a  yard  of  cloth  cost  $4,  how  many 
yards  can  be  purchased  for  §12  ? 

9.  A   jockey,  by    selling    a   horse    for    $45,   gained 
1 -eighth  of  the  cost:  what  was  the  cost? 

10.  If  1  and  1 -third  yards  of  cloth  cost  $5,  how  much 
can  be  purchased  for  $12  ? 

11.  By  selling  5   yards  of   cloth   for   $12,   I  gained 
1 -third  of  the  cost:  what  did  I  pay  per  yard ? 

12.  By  selling   7    yards    of   cloth   for   $21,   I  made 
2-fifths  of  the  cost :  I  paid  for  it  with  wheat,  at  $|  per 
bushel :  how  many  bushels  did  I  give  ? 

13.  If  3-fifths  of  an  apple  cost  2-thirds  of  a  cent,  what 
will  3  apples  cost  ? 

14°.  What  will  be  the  cost  of  11  yards  of  cloth,  if  5 
and  1-half  yards  cost  $4|  ? 

15.  By  selling  cloth  at  $8  a  yard,  1-fifth  of  the  cost 
was  lost :  what  part  would  have  been  gained,  if  the  clotb 
had  been  sold  for  $11  ? 


TABLES.  115 

SECTION  XXIIL— TABLES. 

UNITED  STATES  MONEY. 

10  mills,  marked  m.,  .  .  make  1  cent,  .  .  marked  ct. 

10  cents "      1  dime,  .  "       d. 

10  dimes  or  100  cents.       "      1  dollar,  .  "       $. 

10  dollars "      1  eagle,  .  "       E. 

1.  Repeat  the  table  of  United  States  Money. 

2.  How  many  cents  in  a  half  dime  ?    in  a  quarter  of 
a  dollar  ?     in  a  half  dollar  ? 

3.  In     1  cent    how  many  mills  ?  in     2  ?  in     3  ? 

4.  In     1  dime   how  many  cents  ?  in     2  ?  in     4  ? 

5.  In     1  dollar  how  many  cents  ?  in     2  ?  in     4  ? 

6.  In  20  mills  how  many  cents?  in  30?  in  50? 

7.  In  20  cents  how  many  dimes  ?  in  25  ?  in  30  ? 

8.  At  20  cents  a  yard,  what  will  3  and   3-fourths 
yards  of  calico  cost  ?     how  many  dimes  ? 

9.  At  25  cents  a  yard,  what  will  3  and  1 -third  yards 
r}f  muslin  cost  ?    how  many  dimes  ? 

ENGLISH  MONEY. 

4  farthings  (far.)    .  make  1  penny,  .  .  marked  d. 

12  pence "       1  shilling,  .  "        s. 

20  shillings    .   .   .   .     "       1  pound,  .  .  «       £. 

1.  Repeat  the  table  of  English  Money. 

2.  In  1  penny  how  many  farthings  ?in2?3?     5? 

3.  In  1  shilling  how  many  pence ?      in  2?    3?     4? 

4.  How  many  shillings  in  1  pound  ?    in  2?    3?     4? 

5.  How  many  pence  in  8  farthings  ?     in  12?     16? 

6.  How  many  shillings  in  24  pence  ?     in  36  ?     56  ? 

7.  How  many  pounds  in  40  shillings  ?  in  60  ?    72  ? 


U6  RAY'S   INTELLECTUAL  ARITHMETIC. 

TROY  WEIGHT. 

24  grains  (gr.)    make  1  pennyweight,  .  marked  pwt. 
20  pennyweights      "     1  ounce,  ....        "          oz. 
12  ounces    ...      "     1  pound,.    ...        "          Ib. 

1.  Repeat  the  table  of  Troy  Weight, 

2.  In    1  pennyweight  how  many  grains  ?    in    2?     3? 

3.  In    1  ounce  how  many  pennyweights  ?    in    2  ?     3  ? 

4.  In    1  pound  how  many  ounces  ?  in    3  ?     5  ? 

5.  In  24  oz.  how  many  Ib.?    in  36  ?    48?   28?   56? 

APOTHECARIES  WEIGHT. 
20  grains  (gr.)  .    .  make  1  scruple,     .   .  marked  3. 

3  scruples.    ...       "      1  dram,     ...        "        3- 
8  drams    ....      "      1  ounce,   ...       "        5- 

12  ounces  ....      "      1  pound,  ...       "       Ib. 

1.  Repeat  the  table  of  Apothecaries  Weight. 

2.  In  1  scruple  how  many  grains  ?        in  2  ?     3  ? 

3.  In  1  dram      how  many  scruples  ?    in  3  ?     7  ? 

4.  In  1  ounce     how  many  drams  ?        in  2  ?     3  ? 

5.  In  1  pound    how  many  ounces?       in  3  ?     7? 

6.  In  15  oz.  how  many  Ib.  ?  in  24  ?  in  35  ?  48  ?  50  ? 

AVOIRDUPOIS  WEIGHT. 

16  drams  (dr.).    «    .  make  1  ounce,  .    .  marked  oz. 

16  ounces,     .   .   .    .     "       1  pound, .    .        "  Ib. 

25  pounds,    .    .    .    ,     "       1  quarter,    .        "  qr. 

4  quarters  or  100  Ib.  "       1  hundred  weight,       cwt. 
BO  hundred  weight,      "       1  tun,  ...        "          T. 

1.  Repeat  the  table  of  Avoirdupois  Weight. 

2.  In  2  pounds   how  many  ounces?    in  3?    in  4? 


TABLES. 

3.  In  2  quarters    how  many  pounds  ?    in  3  ?  in  4  ? 

4.  In  2  tuns   how  many  cwt.  ?     in  3  ?    in  4  ?  in  5  ? 

5.  At  2  cents  an  ounce,      what  will  2  Ib.     cost? 

6.  At  2  dollars  a  quarter,  what  will  2  cwt.  cost? 

DRY  MEASURE. 

2  pints  (pt.)  .    .    .  make  1  quart,  .   .    .  marked  qt. 

8  quarts "       1  peck,    ...  "       pk. 

4  pecks "      1  bushel,    .    .  "       bu. 

1.  Repeat  the  table  of  Dry  Measure. 

2.  In  1  quart   how  many  pints ?    in    2?     in 3?      5? 

3.  In  1  peck     how  many  quarts  ?  in    2  ?     jn  4  ?      7  ? 

4.  In  1  bushel  how  many  pecks  ?in2?     in  3?      5? 

5.  In  2  pints    how  many  quarts  ?in5?     in  8?      9? 

6.  In  8  quarts  how  many  pecks  ?    in  16  ?       24  ?    32  ? 

7.  In  4  pecks  how  many  bushels  ?  in  12  ?     in  20  ? 

8.  At  5  cents  a  pt.,  what  will  2  qt.   of  beans  cost  ? 

9.  At 6  cents  a  qt.,  what  will  1  peck  of  corn    cost? 
10.  At  3  cts.  a  qt.,  what  cost  3  and  3-fourths  pk.  of  salt? 

LIQUID  OR  WINE  MEASURE. 

marked    pt. 
"         qt. 
"       gal. 
bl. 

«      hhd. 
"         T. 

1.  Repeat  the  table  of  Wine  Measure. 

2.  In  1  pint      how  many  gills  ?     in  2  ?     in  3  ?     in  5  ? 

3.  In  2  quarts  how  many  pints ?in3?     5?     6?     8? 
2d  Bk.  8 


4    gills  (gi.)  .   .   .  i 
2    pints  

make  1  pint,  .  .  . 
u  1  auart. 

4    quarts    .... 
31-|  gallons   .... 
63    gallons  .... 
4    hogsheads.    .   . 

L         VJ  1*1*  M.   Vj 

"  1  gallon,  .  . 
"  1  barrel,  .  . 
"  1  hogshead,  . 
"  1  tun,  .  .  . 

118  RAY'S   INTELLECTUAL   ARITHMETIC. 

4.  In  1  gal.  how  many  quarts?     in  2  ?  3  ?   7  ?  10? 

5.  In  1  tun   how  many  hogsheads  ?   in  3?   in  5?   8? 

6.  In  4  gi.     how  many  pints  ?     in  7  ?    in  8  ?     in  9  ? 

7.  In  3  pt.     how  many  quarts ?     in  6?    in  8?    11? 

8.  In  4  qt.     how  many  gallons  ?    in  7  ?   in  11  ?    12  ? 

9.  At  12  cents  for  one  pint,  what  will  1  and  3-fourtha 
gallons  of  Port  wine  cost  ? 


LONG  MEASURE. 
12    inches  (in.)  .    .  make  1  foot,  .    .    .  marked    ft. 

3  feet "      1  yard,     .    .         "       yd. 

5£  yards  or  16J  ft.      "      1  rod  (or  pole),    "       rd. 

40    rods  or  220  yd.      "      1  furlong,    .         "     fur. 

8  furlongs    ...      "      1  mile,.    .   .         "      mi. 

1.  Eepeat  the  table  of  Long  Measure. 

2.  In  2  feet  how  many  inches  ?     in  3  ?     in  5  ?     in  8  ? 

3.  In  2  yards  how  many  feet  ?     in  3  ?     in  4  ?     in  5  ? 

4.  In  2  rods  how  many  yards  ?     in  3  ?     in  4  ?     in  5  ? 

5.  In  1  furlong  how  many  rods  ?     how  many  yards  ? 

6.  In  1  mile  how  many  furlongs  ?     in  3?     4?     5? 

7.  How  many  inches  in  8  and  5-sixths  feet? 

8.  In  2  rods  how  many  feet  ?    in  3  rods  ?     4  rods  ? 
5  rods  ?     6  rods  ? 

SQUARE  MEASURE. 

144    square  inches  .  make  1  square  foot,    marked  sq.  ft. 

9  square  feet  .    .      "  1   square  yard,.  "  sq.  yd. 
30J  square  yards   .      "  1  square  rod,   .  "  sq.  rd. 
40    square  rods .    .       "  1  rood,  .    .    .    .  "             R. 

4  roods    ....      "  1  acre,  .   .    .   .  "            A. 
640    acres "  1  square  mile,  .  "  sq.  mi 


TABLES.  119 

1.  Repeat  the  table  of  Square  Measure. 

2.  How  many  square  feet  in  3  square  yards  ?  in    5  ? 

3.  How  many  roods  in  4  acres  ?    in  6  ?    in  8  ?  in  10  ? 


CLOTH   MEASURE. 


4  nails  (na.)  .    .  make  1  quarter,     .    .    marked      qr. 

4  quarters  . 

u 

1  yard,.    .    .    . 

u 

yd. 

3  quarters  . 

(I 

1  ell  Flemish,. 

u 

E.F1. 

5  quarters  . 

u 

1  ell  English,  . 

If 

E.En. 

6  quarters  . 

a 

1  ell  French,  . 

a 

E.Fr. 

1.  Repeat  the  table  of  Cloth  Measure. 

2.  How  many  quarters  in  2  yards  ?     in  3?     4?  5? 

3.  How  many  quarters  in  1  ell  Flemish  ?   in  4  ?  5  ? 

4.  How  many  quarters  in  1  ell  English  ?    in  3  ?  4  ? 

5.  How  many  quarters  in  1  ell  French  ?     in  2  ?  4  ? 

6.  How  much   longer     is  1  ell  French  than  1  ell  Eng- 
lish?    than  1  ell  Flemish? 

7.  There  are  36  inches  in  1  yard :  how  many  inches 
are  there  in  1  quarter?     in  1  nail? 

TIME   MEASURE. 

60  seconds  (sec.) make  1  minute,  marked  min. 

60  minutes "      1  hour,    .  "  hr. 

24  hours "      1  day, .    .  «  da. 

365  days  6  hours  (365J  days)     "      1  year,    .  "  yr. 

100  years "      1  century,  "  cen. 

7  days "      1  week,   .  u  wk. 

4  weeks "      1  month,  "  mon» 

12  calendar  months ....       "      1  year,     .  "  yr. 

NOTE. — One  Solar  year  contains  365  days,  6  hours,  48  minute* 
and  48-  seconds,  or  365 \  days  nearly. 


120  BAY'S   INTELLECTUAL   ARITHMETIC. 

The  following  table  shows  the  names  of  the  different 
months  of  the  year,  and  the  number  of  days  embraced 
in  each. 


January,  1st    mon.,  31  da. 

February,  2d        "      28 

March,  3d       «      31 

April,  4th      «      30 

May,  5th      «      31 

June,  6th      «      30 


July,  7th  mon.,  31  da. 

August,  8th      "  31    « 

September,  9th      "  30    " 

October,  10th      "  31    « 

November,  llth      "  30    " 

December,  12th      "  31    " 


The  number  of  days  in  each  month  of  the  year  may 
be  retained  in  the  mind  by  committing  the  following  lines 
to  memory : 

Thirty  days  has  September, 
April,  June,  and  November; 
Other  months  have  thirty-on-e, 
Except  the  second  month  alone; 
To  this  we  twenty-eight  assign, 
Till  leap-year  gives  it  twenty-nine. 

1.  Repeat  the  table.     "What  make  minutes  ?     days  ? 

2.  In  14  days  how  many  weeks  ?     in  21  ?     in  28  ? 

3.  How  many  minutes  in  1  hour  and  a  quarter  ? 

4.  There   are   12   months   in  1  year :  what  part  of  a 
year  is  1  month  ?     what  part  are  2  months  ?     3  months  ? 
i  months?     5?    6?    8?    9?     10? 

5.  If  there  are  30  days  in  1  month,  what  part  of  a 
month   are   3  days?     what  part  are  5  days?     6   days?, 
10  days?     12   days?     15  days?     18    days?     20   days? 
84  days?    27  days? 


GENERAL    REVIEW. 

APPLICATIONS  OF  MENTAL  ARITHMETIC, 

FOE  ADVANCED  CLASSES. 

SECTION    XXIV. 

THM  preceding  sections  contain  all  the  elementary  forms  of 
Anatysiw.  Those  who  are  properly  acquainted  with  them,  will 
find  bnt  little  difficulty  in  these  applications. 

LESSON  I.— ADDITION. 

SUGGESTION. — Where  the  numbers  are  large,  it  is  better  to  add 
by  10' s  or  100's,  as  the  case  may  be. 

To  add  25  ai*d  35 :  20  and  80  are  50 ;  5  and  5  are  10 ;  50  and 
10  are  60.  Or,  £5  and  30  are  55;  55  and  5  are  60. 

OP    THE    SIGNS. 

The  sign  +,  called  plus,  means  more.  The  numbers 
between  which  it  is  placed  are  to  be  added. 

Thus,  2+4,  (2  plus  4),  shows  that  2  and  4  are  to  be 
added  together. 

The  sign  of  equality,  =,  shows  that  the  number  of 
units  on  the  right  and  left  of  it  are  equal  to  each  other. 

Thus,  2  +  4  =  6,  means  that  2  added  to  4  equal  6  j  and 
is  read,  2  and  4  are  6,  or  2  plus  4  equal  6. 

EXAMPLES.          4  +  2  =  how  many  ? 

3  +  5  =  how  many  ? 

2  +  3  +  4  =  how  many  ? 

1  +  3  +  5  +  7  =  how  many  ? 

$4  +  $3  +  $1  +  $2  =  how  many  $'s  ? 

(121) 


122  KAY'S   INTELLECTUAL   ARITHMETIC. 

1.  What  is  the  sum  of    4+    5+    6+7? 

2.  What  is  the  sum  of    9  +  10  +  11  +    8  ? 

3.  What  is  the  sum  of  20  +  12  +    9  +  11  ? 

4.  What  is  the  sum  of  24  +  20  +  12  +  30  ? 

5.  What  is  the  sum  of  35  +  40  +  15  +  20  ? 

6.  What  is  the  sum  of  50  +  60  +  70  +  80  ? 

7.  What  is  the  sum  of  54  +  20  +  13  +  12  ? 

8.  What  is  the  sum  of  $21+$16+$13+$20? 

9.  Bought  at  one  time  33  gal.  of  oil,  at  another  20, 
at  another  40,  at  another  50,  and  at  another  62 :  how 
many  gal.  did  I  buy  ? 

10.  A  lady  paid  §23  for  a  dress,  $18  for  a  shawl,  and 
69  for  a  bonnet :  how  much  did  she  spend  ? 

11.  I  owe  A   850,   B  $75,  C  $40,  and  D  $20:    how 
much  money  do  I  owe  ? 

12.  I  collected  of  one  man  $110,  of  another  $90,  of 
another  $75,  and  of  another  $50 :  how  much  in  all  ? 

13.  Thirty-two  plus  16  +  20  +  21  +  18  =  what  ? 

14.  Fifty-nine    plus  21  +  32  +  15  +  11  =  what? 


LESSON  II.— SUBTRACTION. 

The  sign  — ,  called  minus,  means  less.  Placed  between 
two  numbers,  it  shows  that  the  one  on  the  right,  is  to  be 
taken  (subtracted),  from  the  one  on  the  left. 

Thus,  4  —  2=2;  read>  4  minus  (less)  2  equal  2. 

EXAMPLES.  5  —    2  =  how  many  ? 

3  +  9  —    8  =  how  many  ? 

1  +  5  +  9  —  10=  how  many  ? 

2  +  3  +  4+8  —  11=  how  many ? 

$5  +  $9  +  $7  +  $4  —  $12  =  how  many  $'s  ? 


RE  VIEW.— MULTIPLICATION.  1 23 

1.  What   number   equals  75   less  40  ?      160  —  120  ? 
100  —  45?     110  —  90?     120  —  95? 

2.  A  boy  having  75  ct,  purchased  55  ct.  worth  of 
goods  :  how  much  change  did  he  receive  ? 

3.  Having  $92,  I  purchased  a  watch  for  $73:  how 
much  had  I  left? 

4.  Bought  a  horse  for  $110,  and  sold  him  for  $145: 
how  much  did  I  make? 

5.  George  bought  candles  for  25  ct.,  soap  for  10  ct., 
sugar  for  35  ct.,  and  starch  for  3  ct. :    he  gave  $1,  and 
received  30  ct.  change  :  was  this  correct  ? 

6.  A  boy  had  $5,  from  which  he  took  at  one  time  $1 
and  50  ct. ;  at  another,  40  ct. ;  at  another,  $1  and  10  ct.: 
how  much  had  he  left  ? 

7.  Ten  plus  22  +  19  less  8  +  3  +  9  +  6=what  ? 

8.  42  +  19  +  13  +  15  —  12  —  17  —  20  —  4=? 

9.  $37  +  $33  +  $45  +  $25  —  $35  —$20  —  $40=  ? 

10.  $125  +  $140  +  $20  —  $100  —  $50— $8— $30= ? 

11.  $160 +  $80 +  130— $210— $30— $10—  $40==? 


LESSON  III.— MULTIPLICATION. 

The  sign  X,  denotes  multiplication,  read,  multiplied 
ly.  When  placed  between  two  numbers,  it  shows  that 
ihey  are  to  be  multiplied  together. 

Thus,  3x5,  means  3  multiplied  by  5. 

EXAMPLES.  3X5  =  how  many  ? 

2X5X7  =  how  many  ? 
2X4X6X3  =  how  many  ? 
$8  +  $12  —  $7  X  10  =  how  many  $'s  ? 


124  RAY'S  INTELLECTUAL  ARITHMETIC. 

1.  What  is  the  product  of  2+6X2?    25X2?    16X3? 
20X5?    22X6?     40X5?     38X3?      60X4?    45x5? 
24X6?    53X9?     65X8? 

2.  What    is    the    product    of    14x6?      4X7X5? 
5X6x7?     9X10X5?     6x8X5? 

3.  What  will  be  the   cost  of  5  yd.  of  cloth,  at  $2 
and  50  ct.  a  yd.  ? 

4.  A  man  traveling  at  the  rate  of  5  mi.  an  hr.,  meets 
a  stage  going  at  the  rate  of  9  mi.  an  hr. :  how  far  from 
the  man  will  the  stage  be  in  10  hr.  ? 

5.  What  cost  9  boxes,  at  7  dimes  each  ? 

6.  What  cost  8  Ib.  and  4  oz.  of  sugar  at  12  ct.  a  Ib.  ? 

7.  What  cost  75  ft.  of  lumber,  at  3  ct.  a  ft.  ? 

8.  If  1  A.  of  land  produce  85  bu.  of  corn,  how  many 
bu.  will  11  A.  produce  ? 

9.  Bought  15  Ib.  of  coffee  at  10  ct.  a  Ib.,  and  13  Ib. 
at  9  ct.  a  Ib,  :  what  did  the  whole  cost  ? 

10.  Henry  has  19  ct ;  George  3  times  as  many,  lack- 
ing 10  :  how  many  have  beth? 

11.  How  many  yd.  in  3  bales  of  cloth,  each  containing 
6  pieces  of  35  yd.  each  ? 


LESSON   IV.— DIVISION. 

The  sign  -7-,  is  read,  divided  ~by.  When  placed  be- 
tween two  numbers,  it  shows  that  the  first  is  to  be  di- 
vided by  the  second. 

Thus,  6-7-2,  means  that  6  is  to  be  divided  by  2. 


EXAMPLES.  6- 

4X5 

4  +  10X  2 
$3  + $12  —  $5X4 


2  =  how  many  ? 

•  2  =  how  many  ? 

•  7  =  how  many  ? 

•  8  =  how  many  $' 


REVIEW.— DIVISION.  1 25 

1.  How  often  is  3  contained  in  48  ?    in  51  ?     in  60  ? 
In  75  ?     in  81  ?     in  90  ?     in  144  ? 

2.  Divide  125  by  5;  multiply  the  result  by  10;  the** 
divide  by  2  :  what  is  the  last  quotient  ? 

3.  Multiply  14  by  20,  and  divide  the  product  by  7* 

4.  What  does  12  X  13  -=-  2  -f-  6  =  ? 

5.  What  does  15  X  12  -*-  3  -f-  12  —  3  ==  ? 

6.  What  does  27  +  9  -f- 12  +  20  —  17  -f-2  X  5  «=? 

7.  What  is    1-half  of  28  +  1-third  of  72  ? 

8.  What  are  9-fifteenths  of  120  +  §  of  60  ? 

9o  If  a  boat  sail  48  mi.  in  12  hr.,  how  far  will  she 
sail  in  6  hr.  ? 

10.  At  15  ct.  a  lb.,  what  quantity  of  beef  can  be  pur- 
chased for  $6  ? 

11.  Three  men  bought  a  horse  for  $90 :  after  keeping 
him  6  wk.j  at  $3  a  wk.,  they  sold  him  for  $99 :  what  did 
each  man  lose  ? 

12.  Seven  multiplied  by  9,  divided  by  3,  2  added,  13 
subtracted,  and  divided  by  5,  will  =  what  ? 

13.  Add  10  to  12  X  3-7-6  +  5-7-8+10  X  5-7-6-^ 
2  =  what  ? 

14.  Seventeen  +  6  —  8  -r-  3  X  8  —  6  +  14  -r-  4  X  8~ 
8+12 --10 +  6  + 7—  12X  6=  what? 

15.  What  number  added  to  itself  will  give  a  sum  equal 
to  14? 

Explanation. — If  a  number  be  added  to  itself,  the  sum   will 
be  2  times  the  number :  14,  then,  is  2  times  what  number  ? 

16.  What  number  added  to  itself  3  times,  will  make  32  ? 

17.  Divide  16  into  2  parts,   so  that  the  second   pari 
will  be  3  times  the  first. 

Explanation. — The  sum  of  the  parts  (16),  will  be   4  time*  the 
first  part 


126  RAY'S    INTELLECTUAL   ARITHMETIC. 

18.  Divide  48  into  2  such  parts  that  the  second  shall 
be  7  times  the  first.     48  is  8  times  what  number  ? 

19.  Divide  24  into   3   parts,   so  that  the  second  shall 
be  2  times  the  first,  and  the  third  3  times. 

20.  A  boy  being  asked  the  number  of  ct.  he  had,  re- 
plied :    "  Five  times  the   number  I  have,  is  just  40  less 
than  10  times  the  number :"  how  many  had  he  ? 

21.  Find  a  number  which,  being  multiplied  by  2 -7- 8 
X  3  —  9-f-3X  3  +  11  X  3  —  1-5-10,  equals  5. 

Explanation. — Begin  with  the  last  number  mentioned,  and  re- 
verse every  operation  indicated  by  the  signs :  thus,  5  X  10  =  50 ; 
60+1  =  51;  51 -J- 3  =  17,  &c. 

22.  What   does  12X5  +  3-r-7+ll-f-5  — 1  +  10 
+  14  .^3  +  19 +  8  —  9  +  17  +  8=? 

23.  What  does  13  +  27  +  14  +  10  -*-  8  +  21  +  13  -*- 
7  +  14  +  20  +  23  +  3  -f- 11  =  ? 

24.  What   does   19  +  2  —  13x6-f-4  +  7  —  12X5 
--7  +  15  —  11  X  8 --12 +  15  — 14=? 


LESSON   V.— PRINCIPLES. 

1.  When  10  was  taken  from  a  number,  only  2-thirds 
of  the  number  remained  :  what  was  the  number? 

2.  The  sum  of  two  numbers  is  12 ;  if  6  be  added  to 
the  sum,  the  result  will  be  twice  the  greater  number  :  what 
are  the  numbers? 

3.  The  sum    of  two   numbers  is   16  more  than  their 
difference :  if  their  difference  be  4,  and  8  one  of  the  num- 
bers, what  is  the  other  number  ? 

4°. The  sum  of  two  numbers  diminished  by  the  less 
gives  15 :  if  10  is  the  less  number,  what  is  their  difference  ? 


REVIEW.— PRINCIPLES.  127 

5.  If  6  be  taken  from  the  difference  of  two  numbers, 
the  remainder  will  be  2 :    if  4  is  one  of  the  numbers, 
what  is  the  other  ? 

6.  If  10  be  added  to  the  difference  of  two  numbers, 
the  sum  will  be  6  more  than  the  greater  number,  which 
is  19  :  what  is  the  less  number  ? 

7.  If  10  be  taken  from  the  sum  of  two  numbers,  of 
which  5  is  one,  there  will  be  8   left:    what  is  the  other 
number  ? 

8.  By  what  part  of  6  must  4  be  X  to  =  f  of  20  ?' 

9.  What  number  -=-  8  will  give  13  for  a  quotient  ? 

10.  What  number  X  12,  will  give  156  for  a  product  ? 

11.  If  15  be  multiplied  by  some  number  and  20  added 
to  the  product,  the  sum  will  be  200 :  what  is  the  multi- 
plier ? 

12.  A  certain  number  X  2,  gives  a  result  as  much  less 
than  20  as  the  number  is  greater  than  7 ;  but  when  it  is 
subtracted   from  11,  it   leaves  the  same  remainder  as  5 
from  7  :  what  is  the  number  ?     why  ? 

13.  Six  is  contained  in  a  certain  number  12  times,  with 
a  remainder  of  5  :  what  is  the  number  ? 

14.  If  12  be  added  to  a  certain  number,  7  will  be  con- 
tained 9  times  in  the  sum,  with  a  remainder  of  1 :  what 
Is  the  number? 

15.  If  13  be  taken  from  a  certain  number,  8  will  be 
contained  10  times  in  the  difference,  with  a  remainder  of 
3:  what  is  the  number  ? 

16.  When  the  divisor  of  132  was  increased  by  6,  the 
quotient  was  found  to  be  11 :  what  was  the  divisor? 

17.  When  the  divisor  of  72  was  multiplied  by  2,  the 
quotient  was  9  :  what  was  the  divisor  ? 

18.  When  the  divisor  of  84  was  divided  by  3,  the  quo- 
tient was  4  :  what  was  the  divisor  ? 


128  RAY'S    INTELLECTUAL   ARITHMETIC. 

19.  If  1  be  added  to  the   number  of  times  a  certain 
number  is  contained  in  60,  the  result  will  be  11  :  what  is 
that  number  ? 

20.  Twice   the  greater   of   two   numbers,  —  2  =  their 
sum,  which  is  20  :  what  are  the  numbers  ? 

21.  A  boy  received  of  his  father  3  ct. ;  of  his  mother 
twice  as  many  less  1  ;  if  he  had  received  from  his  father 
5  ct.  more,  his  father  would  have  given  him  4  times  as 
many  as  his  sister :  how  many  ct.  did  he  receive  ? 


SECTION  XXV.— QUESTIONS. 

LESSON  I. 

1.  If  I  of  a  yd.  of  cloth  cost  $2,  what  cost  ^  of  a  yd.  ? 

2.  If  |  of  a  yd.  of  cloth  cost  $5,  what  cost  |  of  a  yd.  ? 

ANALYSIS. — \-tliird  of  a  yd.  will  cost  \-Tialf  as  much  as 
2-thirds ;  and  3-thirds,  or  a  whole  yd.,  will  cost  3  times  as 
much  as  I-third ;  and, 

One-fourth  of  a  yd.  will  cost  l-fourth  as  much  as  1  yd., 
and  ^-fourths,  3  times  as  much  as  l-fourth. 

3.  If  |  of  a  bl.  of  flour  cost  §3,  what  cost  f  of  a  bl.  ? 

4.  If  4  of  a  yd.  of  muslin  cost  24  ct.,  what  will  y\  of 
a  yd.  cost  ? 

5.  If  |  of  a  tun  of  hay  cost  $15,  what  will  one-half  a 
tun  cost? 

6.  If  |  of  an  orchard  contain  30  fruit  trees,  how  many 
trees  are  there  in  T7^  of  it  ? 

7.  If  1|  yd.  of  cloth  cost  $14,  what  cost  2^  yd.  ? 

8.  If  14,  bl.  of  flour  cost  $5J,  what  cost  2J  bl.  ? 

9.  If  3J  Ib.  of  cheese  cost  20  ct.,  what  cost  2|  Ib.  ? 


RE  VIE  W.— QUESTIONS.  1 29 

10.  A  traveled  30  mi.  in  3f  hr.  :  at  that  rate,  how  far 
can  he  travel  in  7J  hr.  ? 

11.  If  a  man  earn  $1J  in  10  hr.,  how   much  can  he 
earn  in  11  hr.  ? 

12.  A  can  earn  $9?  in  6  da.,  of  8  hr.  each :  how  much 
tan  he  earn  in  7  da.,  of  9  hr.  each  ? 

13.  If  5|  bu.  of  wheat  cost  $94,  what  cost  3|  bu.  ? 

14.  If  8J  is  f  of  a  number,  what  is  f  of  it? 

15.  If  3^  is  2 §  times  some  number,  what  is  2J  times 
that  number  ? 

16.  If  |  of  a  bl.  of  flour  cost  $4i,  what  cost  f  of  a  bl.? 

17.  If  f  of  a  yd.  of  lace  cost  $f ,  what  cost  |  of  a  yd.  ? 

18.  If  the  wages  of  3  men  for  5  da,  is  $30,  what  will 
be  the  wages  of  4  men  for  7  da.  ? 

SUGGESTION. — First  find  the  wages  of  one  man  for  one  day. 

19.  If  6  persons  spend  $36  in  8  da.,  how  much,  at  that 
rate,  would  5  men  spend  in  12  da.  ? 

20.  If  3  men  can  build  12  rd.  of  wall  in  8  da.,  how 
many  rd.  can  5  men  build  in  3  da.  ? 

21.  If  6  horses  eat  36  bu.  of  oats  in  10  da.,  how  many 
bu.  will  5  horses  eat  in  9  da.  ? 

22.  If  5  oxen  eat  2  A.  of  grass  in  6  da.,  in  how  many 
da.  will  12  oxen  eat  8  A.  ? 

23.  If  a  family  of  8  persons   spend   $400  in  5  mon., 
how  much  would  maintain  them  8  mon.,  if  3  more  per- 
sons were  added  ? 

ANALYSIS.— $400  for  5  mon.  is  $80  for  1  mon.  :  if  8  per- 
sons spend  $80  in  1  mon.,  I  person  spends  $10  in  I  mon. 
Hence,  1 1  persons  spend  $1 10  in  1  mow.,  and  $880  in  8  mon. 

24.  If  10  oxen  can  be  kept  on  5  A.  for  3  mon.,  how 
many  sheep  can  be  kept  on  15  A.  for  5  mon.,  if  7  sheep 
eat  as  much  as  1  ox  ? 


130  RAY'S   INTELLECTUAL   ARITHMETIC. 


LESSON    II. 

1.  If  5  men  can  do  a  piece  of  work  in  18  da.,  how 
many  men  will  do  it  in  9  da.  ? 

2.  If  8  men  can  do  a  piece  of  work  in  15  da.,  how 
many  men  can  do  it  in  12  da.  ? 

3.  If  9  pipes   fill  a   cistern  in  2^-hr.,  in  what  time 
will  5  such  pipes  fill  it  ? 

4.  If  5  men  do  a  piece  of  work  in  6  da.,  how  many 
can  do  a  piece  twice  as  large,  in  1 -fifth  the  time? 

5.  If  8  men  can  do  a  piece  of  work  in  5  da.,  in  what 
time  can  5  men  do  it  ? 

6.  If  6  men  can  do  a  piece  of  work  in  5  da.,  in  what 
time  can  they  do  it,  if  they  receive  the  assistance  of  3 
additional  men  when  the  work  is  half  completed  ? 

7.  If  7  men  can  do  a  piece  of  work  in  4  da.,  in  what 
time  can  it  be  done,  if  3  of  the  men  leave  when  the  work 
is  half  completed  ? 

'8.  If  20  Ib.  of  flour  afford  8  five  ct.  loaves,  how  many 
one  ct.  loaves  will  it  furnish  ?  how  many  four  ct.  loaves  ? 
how  many  ten  ct.  loaves  ? 

9.  If  10  Ib.  of  flour  afford  6  five  ct.  loaves,  how 
many  3  ct.  loaves  will  it  furnish  ? 

ANALYSIS. — It  will  afford  5  times  as  many  1  ct.  loaves  as  5 
ct.  loaves :  l-third  as  many  3  ct.  loaves  as  1  ct.  loaves. 

10.  If  a  sack  of  flour  make  20  three  ct.  loaves,  how 
many  4  ct.  loaves  will  it  make  ?     5  ct.  loaves  ? 

11.  If  the  5  ct.  loaf  weigh  8  oz.  when  flour  is  $3  a 
bl.,  what  should  it  weigh  when  flour  is  §1  a  bl.  ?     what  if 
flour  is  §2  a  bl.  ?     $4  a  bl.  ? 

12.  A  4  ct.  loaf  weighs  10  oz.  when  flour  is  $6  a  bl. ; 
what  will  it  weigh  when  flour  is  $5  a  bl.  ? 


RE  VIE  W.— QUESTIONS.  1 3 1 

ANALYSIS. — If  flour  were  $1  a  bl,  it  ought  to  weigh  6  times 
as  much  as  when  flour  is  $6  a  bl. ;  that  is,  60  oz. :  and. 

When  flour  is  $5  a  bl.,  it  ought  to  weigh  J  as  much  as  when 
it  is  $1  a  bl.,  that  is,  12  oz. 

13.  If  the  3  ct.  loaf  weigh  7  oz.  when  flour  is  $3J  a  bl., 
what  ought  it  to  weigh  when  flour  is  $2J  a  bl.  ? 

14.  If   6  men  can  mow  a  field  in  5J  da.,  how  muck 
time  would  be  saved  by  employing  4  more  men  ? 


LESSON   III. 

1.  A  and  B  hired  a  pasture  for  $45 :   A  pastured  4 
cows,  and  B  5  cows  :  what  should  each  pay  ? 

ANALYSIS. — They  together  pastured  9  cows,  of  which  ^  were 
As,  and  f  J3's;  hence,  A  should  pay  |  of  $45,  which  are 
$20;  and  B  f  of  $45,  which  are  $25. 

2.  William  had  3  ct.,  Thomas  4  ct.,  and  John  5  ct. ; 
they  bought  36  peaches  :  what  was  the  share  of  each  ? 

3.  Two  men  paid  $3  for  7^  dozen  oysters  :  the  first 
paid  $2,  and  the  second  $1 :  how  many  should  each  have  ? 

4.  A  and  B   bought  a  horse  for  $40;    A  paid  $25, 
and  B  the  rest :  they  sold  him  for  $56  :  what  should  each 
receive  ? 

5.  A  boat  worth  $860,  of  which  ^  belonged  to  A,  J  to 
*  B,  and  the  rest  to  C,  was  entirely    lost :    what  loss  will 

each  sustain,  it  having  been  insured  for  $500  ? 

6.  Two  boys  bought  a  silver  watch  for  $7 :    the  first 
paid  $2^ ;  the  second,  $4A  ;   they  sold  it  for  $21 :  what 
was  each  one's  share  ? 

7.  A  man  failing,  paid  80  ct.  on  each  dollar  of  his 
indebtedness :  what  did  I  receive,  if  he  owed  me  $60  ? 

ANALYSIS. — 80  ct.  are  f  of  $1 ;  he  therefore  paid  me  |  of 

">,or$48. 


132  KAY'S  INTELLECTUAL  ARITHMETIC 

8.  A  grocer  failing,  pays  60  ct.  on  the  dollar :  what 
mil  B  receive  to  whom  he  owes  $25  ? 

9.  A  trader  failing,  pays  only  15  ct.   on  the  dollar: 
what  will  C  receive  to  whom  he  owes  $80  ? 

10.  A  and  B  rent  a  pasture  for  $25 :    A  puts  in  27 
oxen,  and  B  180  sheep :  what  should  each  pay,  supposing 
an  ox  to  eat  as  much  as  10  sheep  ? 

11.  A  and  B   rent  a  pasture  for  $60 :    A  puts  in  14 
horses,  and  B  15  cows :  what  should  each  pay,  if  2  horses 
eat  as  much  as  3  cows  ? 

12.  A   and  B  rent  a  pasture  for  $75 :    A  puts  in  8 
horses ;  B  15  oxen  and  120  sheep  :  what  should  each  pay, 
if  a  horse  eat  as  much  as  20  sheep,  and  2  horses  as  much 
as  3  oxen  ? 

13.  A  and  B  rent  a  pasture  for  $35  ;  A  puts  in  4  horses 

2  wk. ;  B,  3  horses  4  wk. :  what  ought  each  to  pay  ? 

ANALYSIS. — 4   horses  for  2  wk.  =  1  horse  for  8  wk. ;  and 

3  horses  for  4  wk.  —  1  horse  for  12  wk.  :  8  wk.  and  12  wk. 
are  20  wk ;  hence, 

A  must  pay  ^  or  f  of  the  rent,  =  $l4;  and  B  £§,  or  f  of 
the  rent,  =  $21. 

14.  C  and  D  join  their  stocks  in  trade ;  0  puts  in  $50 
for  4  mon.,  and  D  $60  for  5  mon. :  they  gain  $45 :  what 
is  the  share  of  each? 

15.  Two  masons,  A  and  B,   built  a  wall  for  $81 ;  A 
sent  3  men  for  4  da.,  and  B  5  men  for  3  da. :  what  ought 
each  to  receive  ? 

16.  A  and  B  traded  in  company ;  A  put  in  $2  as  often 
as  B  put  in  $3 ;  A's  money  was  employed  5  mon.,  and 
B's  4  mon. ;  they  gained  $55  :  what  was  each  man's  share  ? 

17.  E  and  F  rented  a  field  for  $27 ;  E  put  in  4  horses 
for  5  mon.,  and  F  10  cows  for  6  mon. :  what  ought  each 
to  pay,  if  2  horses  eat  as  much  as  3  cows  ? 


EEVIEW.— QUESTIONS.  133 


LESSON  IV. 

1.  Divide  20  apples  between  A  and  B,  so  that  A  may 
as  often  as  B  gets  3. 


ANALYSIS.—  Of  each  5  apples,  A  must  get  2  and  B  3J 
In  20  apples  there  are  4  times  5  apples  ;  hence, 

A  must  get  4  times  2  apples,  or  8  apples;  and  B  4  times  3 
zpples,  or  12  apples. 

2.  Divide    28  ct.    between  John  and  James,  so  that 
John  may  get  3  as  often  as  James  gets  4. 

3.  Divide  45  ct.  between  A,  B,  and  C,  so  that  A  may 
get  4  ct.  as  often  as  B  gets  3,  and  0  gets  2. 

4.  In  an  orchard  of  96  trees,  there  are  5  apple-trees 
for  3  peach-trees  :  how  many  of  each  kind  ? 

5.  On  a  farm  there  are  60  animals  —  horses,  cows,  and 
sheep  ;  for  each  horse  there  are  3  cows,  and  for  each  cow 
there  are  2  sheep  :  how  many  animals  of  each  kind  ? 

6.  A  school  of  35  pupils  has  2  boys  for  3  girls  :  how 
many  of  each  in  the  school  ? 

7.  What  number  is  that  which  being  added  to  3  times 
itself  will  make  48  ? 

8.  Divide  42  plums  between  A,  B,  and  C,  so  that  B 
may  get  twice,  and  C  three  times  as  many  as  A. 

9.  Mary  has  25  yd.  of  ribbon  :    she  wishes  to  divide 
it  into  two  parts,  so  that  one  shall  be  4  times  the  length 
of  the  other  :  what  will  be  the  length  of  each  part  1 

10.  Divide  35   cherries   between    Emma,    Agnes,    and 
Sarah,  so  that  Agnes  shall  have  twice  as  many  as  Emma, 
and  Sarah  twice  as  many  as  Agnes. 

11.  Divide  28  into  two  parts,  so  that  one  shall  be  3 
fcimes  a  certain  number,  and  the  other  4  times. 

2d  Bk.         9 


134  RAY'S   INTELLECTUAL   ARITHMETIC. 


LESSON   V. 

1.  What  part  of  8  is  2?     what  part  is  4?     is  1  ? 

2.  How  many  times  does  10  contain  2  ?     2  is  what 
part  of  10? 

3.  Twelve  is  how  many  times  2  ?     2  is  what  part  of 

12  ?     what  is  the  ratio  of  2  to  12  ? 

EXPLANATION. — When  two  numbers  are  compared,  to 
see  how  many  times  greater  one  is  than  the  other,  what 
do  you  find?  Ans.  The  ratio  of  the  numbers. 

How  do  you  find  the  ratio  or  relation  of  two  numbers? 
Ans.  Divide  the  second  number  by  the  first. 

4.  How  many  times  does  18  contain  9  ?     what  is  the 
ratio  of  9  to  18? 

5.  What  is  the  ratio  of  12  to  36  ?     9  to  45  ?     11  to  66? 

13  to  52  ?    2  to  1  ?    4  to  3  ? 

6.  What  is  the  ratio  of  2i  to  5  ?     6]  to  12-^  ?     \  to  \  ? 
of  \  to  §  ?    |  to •  i  ?     i  to  f  ? 

is  b  o  O  xv  o 

7.  If  the  ratio  of  two  numbers  is  5,  and  6  is  the  less 
number,  what  is  the  greater  ? 

8.  The  ratio  of  7  to  21  is  equal  to  the  ratio  of  some 
number  to  36  :  what  is  the  number  ? 

9.  Five  less  than  the  ratio  of  2  to  20  is  |  of  the  ratio 
of  2  to  what  ? 

10.  The  ratio    of  2  to  18,  plus  3,  is   7  less  than  the 
ratio  of  2  to  what? 

11.  The  ratio  of  9  to  27,  increased  by  5,  is  equal  to 
the  ratio  of  2 i  to  what? 

12.  Divide  25  ct.  between  John  and   George,   so  that 
their  shares  shall  be  in  the  ratio  of  2  to  3. 

Explanation. — When  two  numbers  are  in  the  ratio  of  2  to  3, 
one  will  contain  2  as  often  as  the  other  contains  3. 


REVIEW.— QUESTIONS.  135 

13.  Divide  the  number  48  into  two  parts  that  shall  be 
in  the  ratio  of  5  to  7. 

14.  Divide  the  number  60  into  three  parts  that  shall 
be  to  each  other  as  3,  4,  and  5. 

15.  Divide  the  number  70  into  4  parts  that  shall  be 
to  each  other  as  1,  2,  3,  and  4. 

16.  Divide  the  number  22  into  two  parts  that  shall  be 
to  each  other  as  2|  to  3. 

ANALYSIS. — 2^  and  3  are  5J  units.  The  first  part  will 
therefore  be  as  many  times  2^,  and  the  second  as  many  times 
3,  as  5J  are  contained  times  in  22. 

17.  Divide   16   apples  between   Henry  and  Oliver,  so 
that  their  shares  shall  be  in  the  ratio  of  1J  to  2J. 

18.  Divide  the  number  39  into  three  parts  that  shall 
be  to  each  other  as  ^,  J-,  and  J. 

19.  Divide   14  ct.   between  A  and  B,  so  that  B  may 
have  1J  times  as  many  as  A. 

ANALYSIS. — As  often  as  A  gets  1  ct.,  B  gets  1J;  that  is,  of 
each  2J  ct.  A  gets  1,  and  B  \\ct. ;  Jience, 

A  gets  as  many  times  1  ct.  and  B  as  many  times  1|  ct., 
as  2J  ct.  are  contained  times  in  14  ct.  Ans.  A  6  ct.,  B  8  ct. 

20.  John  and  James  together  have  33  marbles;  James 
has  If  times  as  many  as  John  :  how  many  has  each  ? 

21.  William's  age  is  1|  times  Frank's  age;  the  sum  of 
their  ages  is  32  yr. :  what  is  the  age  of  each  ? 

22.  A  basket  contains  30  apples  :  the  number  of  those 
which   are   sound,  is   2J  times  the  number  of  those  not 
sound  :  how  many  are  there  of  each  ? 

23.  Two  men  built  27  ft.  of  wall :  how  much  did  eacli 
build,  if  one  built  J  as  much  as  the  other  ? 

24.  A,  B,  and  C,  have  $42 ;   B  has  half  as  many  as 
A,  and  C  half  as  many  as  B :  how  many  has  each  ? 


136  RAY'S   INTELLECTUAL   ARITHMETIC. 

25.  On  a,  farm  there  are  104  animals — hogs,  sheep,  and 
cows  ;  there  are  f  as  many  sheep  as  hogs,  and  |  as  many 
cows  as  sheep :  how  many  are  there  of  each  ? 


LESSON  VI. 

1.  Divide   15  ct.  between  A  and  B,  so  that  B  may 
have  3  more  than  A. 

ANALYSIS. — Reserving  3  ct.  for  the  number  that  B  receives 
more  than  A,  there,  are  12  ct.  left',  dividing  these  equally,  A 
will  get  6  <tf. ;  and  B,  6  ct.  plus  3  ct.  reserved,  or  9  ct. 

2.  Thomas  has  5  apples  more  than  James,  and  they 
both  together  have  19:  how  many  has  each? 

3.  The   sum    of  two  numbers  is  31,  and  the   greater 
exceeds  the  less  by  7  :  what  are  the  numbers  ? 

4.  Thomas  and  James  each  had  the  same  number  of 
ct.,  when   Thomas   found  8  ct.   more ;  they  then  had  to- 
gether 32  ct. :  how  many  had  each  ? 

5.  Thomas  and  William  each  bought  the  same  number 
of  peaches ;  after  Thomas  ate  4,  and  William  6,  they  both 
together  had  20  left :  how  many  had  each  remaining  ? 

6.  Mary  bought  twice  as  many  cherries  as  Sarah ;  and 
after  Mary  ate   7,  and  Sarah  5,  they  had  only  24   left: 
how  many  had  each  left  ? 

7.  If  5  be  added  to  the  treble  of  a  certain  number, 
the  sum  will  be  50  :  what  is  the  number  ? 

8.  If  |  of  a  certain  number  be  increased  by  10,  the 
gum  will  be  31 :  what  is  the  number  ? 

9.  If  |  of  a  number  be  diminished  by  7,  the  remain- 
der will  be  21 :  what  is  the  number  ? 

10.  James  is  4  yr.  older  than  Henry,  and  Henry  is 
E  yr.  younger  than  Oliver  j  the  sum  of  their  ages  is  37  yr. : 
what  is  the  age  of  each  ? 


REVIEW.— QUESTIONS.  137 

11.  Mary  has  8  ct.  more  than  Jane,  and  Sarah  3  less 
than  Mary ;  they  all  have  43  ct. :  how  many  has  each  ? 

12.  The  sum  of  the  ages  of  Mary  and  Frank  is  42  yr.; 
Mary  is  twice  as   old   as  Frank,  less  3  yr. :   what  is  the 
age  of  each  ? 

13.  I   bought  a  watch,  a  chain,  and  a  ring,  for  $62 ; 
the  chain  cost  $5  less  than  the  ring,  and  the  watch  $12 
more  than  the  chain  :  what  did  I  pay  for  each  ? 

14.  Thirty  ct.  are  6  ct.  less  than  £  of  ^  of  my  money : 
how  much  have  I  ? 

15.  John   has   twice  as  much  money  as  James,  +  $3 ; 
Frank  has  as  much  as  John  and  James,  +  $7  ;  together 
they  have  $55  :  how  much  has  each  ? 


LESSON    VII.     • 

1.  Divide  the  number  15  into  two  parts,  so  that  the 
less  part  may  be  |  of  the  greater. 

ANALYSIS. — The  greater  part  being  3-thirds,  and  the  less, 
2-thirds,  their  sum  will  be  5-thirds :  15,  then,  is  5-thirds  of 
what  number  ? 

2.  Thomas  and  John  have  $60  to  pay ;  John  has  |  as 
much  to  pay  as  Thomas  :  what  must  each  pay  ? 

3.  The  time  past  from  noon  is  equal  to  half  the  time 
to  midnight :  what  o'clock  is  it  ? 

4.  The  time   elapsed  since  noon  is  |   of  the  time  to 
midnight :  what  is  the  hour  ? 

5.  I  had  56  mi.  to  travel  in  2  da. ;  the  second  da.,  I 
went  |  as  far  as  the  first :  how  far  did  I  travel  each  da.  ? 

6.  Divide  100  into  two  such  parts,  that  If  of  the  first 
less  8  will  =  the  second. 

7.  Divide  the  number  45  into  three  such  parts,  that 
the  second  shall  be  J,  and  the  third  J  of  the  first  part. 

The  sum  of  all  the  parts  will  be  nine-fourths  of  the  first  part 


138  RAY'S   INTELLECTUAL   AKITHMETIC. 

8.  A,    B,  and   C,  together   have  40  ct. ;    B  has  f  as 
many  as  A,  and  C  f  as  many  as  B  :  how  many  has  each  ? 

9.  A   tree  70  ft.  long  was  broken  into  3  pieces;  the 
middle  part  was  |  of  the  top  part ;  the  lower  part  was  | 
of  the  middle  part :  what  was  the  length  of  each  ? 

10.  I  bought  a  hat,  coat,  and  vest,  for  §34  ;  the   hat 
cost  |  of  the  price  of  the  coat,  and  the  vest  f  the  price 
of  the  hat :  what  was  the  cost  of  each  ? 

11.  In  a  field  containing  55  sheep  and  cows,  A  of  the 
cows  =  §  of  the  sheep  :  how  many  are  there  of  each  ? 

12.  The  sum  of  two  numbers  is  100  ;  and  A  of  the  less 
equals  |  of  the  greater :  what  are  the  numbers  ? 

13.  One-fourth  of  Mary's  age  =  J  of  Sarah's,  and  the 
sum  of  their  ages  is  14  yr. :  what  the  age  of  each  ? 

14.  Divide  38  ct.  between  A  and  B,  so  that  f  of  A's 
share  may  be  equal  to  f  of  B's. 

ANALYSIS. — If  f  of  As  share  =  f  of  B's,  then  i  of  As 
share  =  ^  of  | ;  that  is.  T3Q  of  It's ;  and  the  whole  of  A's  = 
3  times  T30,  that  is,  -^  of  B's ;  and, 

Hence,  {g  of  B's  +  -^  of  B's  =  i§  of  B's  =  38  ct. :  38  ct, 
then,  are  ||  of  what  number  ? 

15.  Divide  the  number  51  into  two  such  parts,  that  |  of 
the  first  will  equal  |  of  the  second. 

16.  In  an  orchard  of  65  apple  and  peach-trees,  f  of  the 
apple-trees  =  4  Of  the  peach-trees :  how  many  are  there 
of  each  ? 

17.  From  C  to  D  is  66  mi. ;  A  left  C  at  the  same  time 
B  left  D ;  when  they  met,  f  of   the  distance  A  had  trav- 
eled =g     of    the    distance     B   had   traveled:    how  much 
farther  did  B  travel  than  A  ? 

18.  The  time  past  noon,  +  3  hr.,  is  equal  to  £  of  the 
time  to  midnight :  what  is  the  hour  ? 


REVIEW.— QUESTIONS.  139 

19.  What  is  the  hour  in  the  afternoon,  when  the  time 
past  noon  is  equal  to  J  of  the  time  past  midnight  ? 

Explanation. — Since  the  time  past  noon  is  one-fifth  of  the  whole 
time  from  midnight,  the  time  from  midnight  to  noon,  which  is  12 
hr.,  must  equal  the  remaining  four-fifths  of  the  time. 

20.  What  is  the  hour  in  the  afternoon,  when  the  time 
past  noon  is  |  of  the  time  past  midnight  ? 

21.  What  is  the  hour  of  the  day,  when  ^  of  the  time 
past  noon  is  ^  of  the  time  past  midnight  ? 


LESSON   VIII. 

1.  What  number  is  that  to  which,  if  its  half  be  added, 
the  sum  will  be  15  ? 

ANALYSIS. —  The  number  +  its  ^  =  f  of  the  number.     Now} 
if  |  —  15,  J—5,  and  f ,  or  the  whole  number,  =  10. 

2.  What  number  is  that  to  which  if  its  f  be  added, 
the  sum  will  be  20? 

3.  If  to  Mary's  age  its  f  be  added,  the  sum  will  be 
21  yr. :  what  is  her  age  ? 

4.  What  number  is  that  which  being  doubled,  and  in- 
creased by  its  |,  the  sum  will  be  52  ? 

5.  What  number  is  that  which  being  doubled,  and 
diminished  by  its  ^,  the  remainder  will  be  40  ? 

6.  What  number   is   that  which   being   trebled,   and 
diminished  by  its  f ,  the  remainder  will  be  48  ? 

7.  If  to  David's  age  you  add  its  ^  and  its  |,  the  sum 
will  be  26  :  what  is  his  age  ? 

8.  If  to   Sarah's  age  you  add  its   |,  its  J-,  +  10  yr., 
the  sum  will  be  twice  her  age  :  how  old  is  she  ? 

9.  Thomas  spent  |  of  his  money,  and  had  30  ct.  left ; 
how  much  had  he  at  first  ? 


140  RAY'S   INTELLECTUAL   ARITHMETIC. 

10.  If  to  a  certain  number  you  add  its  J,  its  |  +  27, 
the  number  will  be  trebled :  what  is  the  number  ? 

11.  A  father  is  40  yr.  older  than  his  son  ;    the  son's 
age  is  T3T  of  the  father's  age :  what  is  the  age  of  each  ? 

12.  If  to  Susan's  age  you  add  its  4  +  18  yr.,  the  sum 
will  be  3  times  her  age  :  how  old  is  she  ? 

13.  A  piece  of  flannel,  losing  |  of  its  length  by  shrink- 
age, measured  28  yd. :  what  was  its  length  ? 

14.  Tha  distance  from  A  to  B  is  ^  the  distance  from 
C  to  D,  and  f  of  the  distance  from  A  to  B,  +  20  mi.,  = 
the  distance  from  C  to  D  :  what  is  the  distance  from  A 
to  B,  and  from  C  to  D  ? 

15.  My  age  +  its  ^,  its  i,  and   its   f  =  94 :   what  is 
my  age  ? 

LESSON  IX. 

1.  If  A  can  do  a  piece  of  work  in  2  da.,  what  part 
of  it  can  he  do  in  1  da.  ? 

2.  A  can  drink  a  keg  of  cider   in  4  da. :  what  part 
of  it  can  he  drink  in  1  da.  ? 

3.  B  can  do  a  piece  of  work  in  4  a  da. :  how  many 
times  the  work  can  he  do  in  1  da.  ? 

4.  C  can  mow  a  certain  lot  in  |  of  a  da. :  how  many 
such  lots  can  he  mow  in  a  da.  ? 

5.  A  can  mow  a  certain  field  in  2^  da. :  what  part  of 
it  can  he  mow  in  1  da.  ? 

ANALYSIS. — If  he  can  mow  the  field  in  2^  da.,  he  could  mow 
twice  the  field  in  5  da.,  and  £  of  twice  the  field,  that  is,  f  of 
ihe  field  in  1  da. 

6.  B  can  dig  a  trench  in  3^-  da. :  what  part  of  it  can 
he  dig  in  1  da,  ? 

7.  C  can  walk  from  Cincinnati  to  Dayton  in  3^  da. : 
what  part  of  the  distance  can  he  walk  in  2  da.  ? 


RE  VIEW.—  QUESTIONS.  141 

8.  A  can  do  -J  of  a  piece  of  work  in  1  da.,  and  B  J  of 
It  :  what  part  of  the  work  can  both  do  in  a  da.  ? 

9.  A  can  do  4,  B  -j,  and  C  I  of  a  piece  of  work  in  1 
da.  :  what  part  of  it  can  they  all  do  in  a  da.  ? 

10.  If  A  can  do  a  piece  of  work  in  2  da.,  and  B  in  3  da.  : 
in  what  time  can  they  both  together  do  it  ? 

ANALYSIS.  —  If  A  does  it  in  2  da.,  he  can  do  J  of  it  in  1  da.  ; 
and  if  B  does  it  in  3  da.,  he  can  do  i  of  it  in  1  da.  ;  hence, 

Both  can  do  «|  +  \  —  f  in  1  da.  :  it  will  require  them  both 
as  many  da.  as  |  are  contained  times  in  1,  (the  whole  work). 
Ana. 


11.  A  can  dig  a  trench  in  6  da.,  and  B  in  12  da.  :  in 
what  time  can  they  both  together  do  it  ? 

12.  C  alone  can  do   a  piece  of  work  in  5  da.,  and  B 
in  7  da.  :  in  what  time  can  both  do  it  ? 

13.  A  can  do  a  piece  of  work  in  2  da.,  B  in  3  da.,  and 
C  in  6  da.  :  in  what  time  can  all  three  do  it  ? 

14.  A  cistern   has  3  pipes;  the  first  will  empty  it  in 
3  hr.,  the  second  in  5  hr.,  and  the  third  in  6  hr.  :  in  what 
time  will  all  three  empty  it  ? 

15.  A  and  B  mow  a  field  in  4  da.;  B  can  mow  it  alone 
in  12  da.  :  in  what  time  can  A  mow  it? 

Explanation.  —  Both  mow  one-fourth  in  1  da.,  and  B  one-twelfth 
in  1  da.;  therefore,  A  can  mow  one-fourth  less  one-twelfth,  which 
is  one-sixth  in  1  da.  ;  hence,  he  can  mow  it  in  6  da. 

16.  A  man  and  his  wife  can  drink  a  keg  of  beer  in  12 
da.  ;  when  the  man  is  away,  it  lasts  the  woman  30  da.  : 
in  what  time  can  the  man  drink  it  alone  ? 

17.  Three  men,  A,  B,  and  C,  can  together  reap  a  field 
of  wheat  in  4  da.  ;  A  can  reap  it  alone  in  8  da.,  and  B 
in  12  da.  :  in  what  time  can  C  reap  it  ? 

18.  A  can  do  a  piece  of  work  in  J  a  da.,  and  B  in  J  of 
a  da.  :  how  long  will  it  take  both  to  do  it  ? 


142  RAY'S  INTELLECTUAL  ARITHMETIC. 

ANALYSIS. — If  A  does  the  work  in  J  of  a  da.,  he  can  do  2 
times  the  work  in  I  da. ;  and 

If  B  does  the  work  in\  of  a  da.,  he  can  do  3  times  the  work 
in  I  da. ;  hence, 

They  both  together  can  do  5  times  the  work  in  1  da.,  or  the 
whole  work  in  ±  of  a  da. 

19.  A  cistern  has  two  pipes ;   by  the  1st  it  may  be 
emptied  in  *  of  an  hr.,  and  by  the  2nd  in  1  of  an  hr. : 
in  what  time  will  it  be  emptied  by  both  together  ? 

20.  A  can  alone  dig  a  cellar  in  2 \  da.,  and  B  in  3J  da.  : 
in  what  time  can  they  both  together  dig  it  ? 

Explanation. — A  digs  2-fifths  in  1  da.,  and  B  3-tenths  in  1  da. ; 
hence,  they  both  dig  7-tenths  in  1  da. 

21.  C  can  reap  a  field  of  wheat  in  5  da.,  and  D  can 
reap  it  in  3J  da, :  in  what  time  can  both  reap  it  ? 

22.  A  can  do  a  piece  of  work  in  3  da,,  and  B  a  piece 
3  times  as  large  in  7  da. :  in  what  time  can  they  together 
do  a  piece  5  times  as  large  as  the  piece  done  in  3  da.  ? 

23.  A  cistern  of  100  gal.,  is  emptied  in  20  min.  by  3 
pipes ;  the  1st  discharges  \  a  gal.  in  a  min. ;  the  2nd,  \\ 
gal. :  how  much  does  the  3rd  discharge  ? 


LESSON  X. 

1.  Two-thirds  of  14  is  f  of  what  number? 

ANALYSIS. — 1  \  is  f ;  f  of  f  is  twice  i  of  f ;  J  of  f  is  f ,  and 
twice  |  are  |;  if  i  is  %  of  some  number,  ±  of  f,  or  §  is  %of 
the  number ;  |  is  %  of  seven  times  |,  which  are  M. 

2.  |  of  5|  is   |   of  what  number  ? 

3.  4  °f  4f  is  j^  of  what  number  ? 

4.  f  of  5|  is  T7n  of  what  number  ? 

5.  |  of  2|  is  \  of  how  many  times  2  ? 


REVIEW.— QUESTIONS.  143 

6.  Three-fifths      of  li  is   f   of  how  many  times  4? 

7.  Three-fourths  of  3J-  is  j   of  how  many  times  3? 

8.  John  has  10  marbles,  and  f-  of  what  John  has  is 
the  T87  of  what  James  has  :  how  many  has  James  ? 

9.  Jane  received  f  of  60  plums ;  she  gave  away  |  of 
her  |  :  how  many  were  left  ? 

10.  James  has  a  given  distance  to  travel ;  after  going 
35  mi.,    there   remains    f  of   the  distance :   when  he  has 
gone  I  of  the  remainder,  how  far  must  he  then  go  ? 

11.  A  horse  cost  $40 ;   f  of  the  price  of  the  horse  =  f  of 
the  price  of  the  cart :  what  did  the  cart  cost  ? 

12.  B's  coat  cost  $27,  and  his  hat  $8;  £  of  the  cost 
of  the  coat  +  f  that  of  the  hat,  =  f  of  the  cost  of  his 
watch  :  what  did  the  watch  cost  ? 

13.  Mary  lost  f  of  her  plums ;  she  gave  f  of  the  re- 
mainder to  Sarah,  and  had  6  plums  left :  how  many  had 
she  at  first  ? 

14.  John  has   12  ct.  ;    f  of  his  money  =  J  of   £   of 
William's  money  :  how  much  has  William  ? 

15.  From  A  to  B  is  36  mi. ;  f  of  this  is  f  of  the  dis- 
tance from  C  to  D :  what  is  the  distance  from  C  to  D  ? 

16.  On   counting  their  money,   it  was  found  that  A 
had  12  ct.  more  than  B  ;  and  that  ^  of  B's  =  f  of  A's  : 
how  much  had  each  ? 

17.  In  an  orchard,  J  are  apple-trees,  J  are  pear-trees, 
TV2  are  plum-trees,  and  the  remainder,  which  is  32,  cherry- 
trees  :  how  many  trees  are  thm*e  of  each  kind  ? 

18.  In  an  orchard  of  apple  and  pear-trees,  the  latter 
are  f  of  the  whole ;  the  apple-trees  are  25  more  than  the 
pear-trees :  how  many  are  there  of  each  ? 

19.  In  an  orchard  of  apple,  plum,  and  cherry-trees,  69 
in  all,   the    plum-trees  =  J    of   the    apple-trees,  and  the 
cherry-trees  =  J   of   the  apple-trees  +  i   of   the  plum  : 
how  many  trees  are  there  of  each  kind  ? 


144  RAY'S    INTELLECTUAL   ARITHMETIC. 

20.  The  age  of  Jane  is  |  of  the  age  of  Sarah,  and  |  of 
both  their  ages  is  |  of  the  age  of  Mary,  which  is  12  yr. ; 
what  are  the  ages  of  Jane  and  Sarah  ? 

21.  How  many  times  T3T  of  55,  is  twice  that  number 
of  which  |  of  30  is  |  ? 

22.  John's  money  is  f  of  Charles's ;  and  |  of  John's 
+  §33  =  Charles's  :  how  much  has  each  ? 


LESSON   XI. 

1.  A  hare  takes  4  leaps  while  a  hound  takes  3 ;  2  of 
the  hound's    leaps  =  3  of  the   hare's :    how   many  leaps 
must  the  hound  take  to  gain  the  length  of  a  hare's  leap 
on  the  hare  ? 

ANALYSIS. — Since  2  of  the  hound's  leaps  —  3  of  the  hare's, 
1  of  the  hound's  leaps  =  l^of  the  hares,  and 3  of  the  hound's 
leaps  —  4^  of  the  hares;  hence. 

In  taking  3  leaps  the  hound  gains  %  the  length  of  a  hare's 
leap  on  the  hare;  therefore, 

The  hound  must  take  6  leaps  to  gain  1  leap  on  the  hare. 

2.  Henry  takes  6  steps  while  John  takes  5 ;  but  4  of 
John's  steps  =  5  of  Henry's  :  how  many  steps  must  John 
take,  to  gain  one  of  Henry's  steps  on  him? 

3.  Henry  is  30  steps  before  John,  but  John  takes  7 
steps  while  Henry  takes  5  :  supposing  the  length  of  their 
steps  to  be   equal,   how   many  steps  must  John  take  to 
overtake  Henry  ? 

4.  A  hare  is  10  leaps  before  a  hound,  and  takes  4  leaps 
while  the  hound  takes  3  ;  but  2  of  the  hound's  leaps  =  3  of 
the    hare's :    how   many    leaps    must   the   hound   take  to 
catch  the  hare  ? 

ANALYSIS. — Since  the  hound  takes  3  leaps  while  the  hare 
takes  4,  the  hound  will  take  1  leap  while  the  hare  takes  1| 
leaps,  and  2  leaps,  while  the  hare  takes  2| ;  but, 


RE  VIE  W.— QUESTIONS.  1 45 

By  the  second  condition,  2  of  the  hounds  leaps  —  3  of  the 
hare's ;  therefore, 

In  making  2  leaps  the  hound  gains  |  of  a  hards  leap  on  the 
hare ;  that  is,  3  leaps  —  2f  leaps ;  hence, 

To  gain  10  of  the  hares  leaps,  the  hound  must  make  as  many 
times  2  leaps  as  ±  is  contained  times  in  1 0,  that  is,  60  leaps. 

5.  A  hare  is  100  leaps  before  a  hound,  and  takes  5 
leaps  while  the  hound  takes  3 ;  but  3  leaps  of  the  hound 
=  10  of  the  hare  :  how  many  leaps  must  the  hound  take 
to  catch  the  hare  ? 

6.  N  is  35  steps  ahead  of  M,  and  takes  7  steps  while 
M  takes  5 ;  but  4  of  M's  steps  =  7  of  N's  :  how  many 
steps  must  M  take  to  overtake  N  ?     How  many  more  will 
N  have  made,  when  he  is  overtaken  ? 

7/*  A  hare  is  8  leaps  before  a  hound,  and  takes  3  leaps 
while  the  hound  takes  2 ;  but  2  of  the  hound's  leaps  are 
equal  to  3  of  the  hare's  :  will  the  hound  catch  the  hare, 
and  if  not,  why  ? 

8.  A  fish's  head  is  6  in.  long ;    its  tail  is  as  long  as 
the  head  and  half  of  its  body ;  and  the  body  is  as  long 
as  both  head  and  tail :  what  is  the  length  of  the  fish  ? 

ANALYSIS. — Since  the  tail  is  as  long  as  the  head  and  half 
the  body,  the  tail  is  6  in.  +  ^  the  body ;  but, 

The  body  =  the  head  and  tail ;  therefore,  the  body  =  6  in.  + 
6  in.  +  ^  the  body ;  that  is, 

The  body  •=  12  in.  +  i  the  body ;  therefore,  J  the  body  is  12 
in.,  and  the  body  is  24  in.  long. 

The  tail  —  6  in.  +  J  of  24  in.  =  18  in.  Therefore,  the  whole 
length  is  6  in.  +  24  in.  +  18  in.  =  48  in. 

9.  A  trout's  head  is  4  in.  long ;  its  tail  is  as  long  as 
its  head  and  |  of  its  body  ;  the  body  is  as  long  as  its 
head  and  tail :  what  is  its  length  ? 

10.  A  has  10  ct. ;  C  has  as  many  as  A,  + 1  as  many 
as  B ;  B  has  as  many  as  both  A  and  C :  how  many  ct. 
have  B  and  C  each  ? 


146  RAY'S   INTELLECTUAL   ARITHMETIC. 

11.  A  man  bought  a  sheep,  cow,  and  horse ;  the  sheep 
cost  $8;  the  cow,  as  much  as  the  sheep,  and  1  as  much 
as  the  horse ;  and  the  horse  cost  twice  as  much  as  both 
the  sheep  and  cow :  what  did  each  cost  ? 

12.  The  head  of  a  fish  weighs  8  Ib. ;  the  tail  weighs  3  Ib. 
more  than  the  head  and  half  the  body ;    and  the   body 
weighs  as  much  as  both  the  head  and  tail :  what  is  the 
weight  of  the  fish  ? 

13.  B  is  pursuing  A,  who  is  some  distance  in  advance ; 
B  goes  4  steps,  while  A  goes  5,  but  3  of  A's  steps  =  2 
of  B's ;  B  goes  30  ft.  before  overtaking  A  :  how  many  ft. 
is  A  in  advance  of  B  ?     Ans.  5  ft. 


LESSON   XII. 

1.  A  gentleman  meeting  some  beggars,  found  that  if 
he  gave  each  of  them  3  ct.,  he  would  have  12  ct.  left,  but 
if  he  gave  each  of  them  5  ct.,  he  would  not  have  money 
enough  by  8  ct. :  how  many  beggars  were  there  ? 

ANALYSIS. — Each  beggar  will  receive  2  ct.  more  when  there 
are  5  ct.  given  to  each,  than  when  there  are  3  ct. ;  but, 

Since  the  money  to  be  distributed  is  12  ct.  more  than  3  ct. 
for  each  beggar,  and  5  ct.  for  each  beggar  is  S  ct.  more  than 
the  money,  it  will,  therefore,  require  20  ct.  more  to  give  5  ct.  to 
each,  than  to  give  3  ct. ;  hence, 

As  each  beggar  gets  2  ct.  more,  it  will  take  as  many  beggars 
to  get  20  ct.  more,  as  2  ct.  are  contained  times  in  20  c£,  which 
are  10.  Ans.  10  beggars. 

2.  A  father  wishes  to  distribute  some  peaches  among 
his  children ;  if  he  gives  each  of  them  2  peaches,  he  will 
have  9  left ;  but  if  he  gives  each  4  peaches,  he  will  have 
3  left :  how  many  children  has  he  ? 

Explanation. — The  difference  between  having  3  peaches  left  ay 
9  left,  is  6. 


RE  VIE  W.— QUESTIONS.  1 47 

3.  Mary  wishes   to  divide  some  cherries  among  her 
playmates  ;  she  finds  that  if  she  gives  each  of  them  5,  she 
will  have  21  left;  but  if  she  gives  each  8,  she  will  have 
none  left :  what  is  the  number  of  her  playmates  ? 

4.  A  lady  wished  to  buy  a  certain  number  of  yd0  of 
silk  for  a  dress ;  if   she  paid  $1  a  yd.,  she  would  have 
$5  left ;  but  if  she  paid  $H  per  yd.,   it  would  take  all 
her  money  :  how  many  yd.  did  she  want  ? 

5.  To  buy  a  certain  number  of  oranges  at  8  ct.  each, 
requires  6  ct.  more  than  all  the  money  James  has ;  but 
if  he  buys  the  same  number  of  lemons,  at  3  ct.  each,  he 
will  have  29  ct.  left :  how  much  money  has  he  ? 

6.  There  are  two  pieces  of  muslin,  each  containing  the 
same  number  of  yd.  ;  to  buy  the  first  at  12^  ct.  a  yd., 
requires  40  ct.  more  than  to  buy  the  second  at  10  ct.  a  yd. : 
how  many  yd.  in  each  ? 

7.  Five  times  a  certain  number  is  16  more  than  3 
times  the  same  number  :  what  is  the  number  ? 

ANALYSIS. — 5  times  any  number  less  3  times  the  same  num- 
ber, is  2  times  the  number;  therefore,  2  times  the  number  is  16. 

'  8.  Thomas's  age  is  three  times  that  of  James,  and  the 
difference  of  their  ages  is  10  yr. :  what  is  the  age  of  each  ? 

9.  The  age  of  A  is  5  times  the  age  of  B ;  and  the 
age  of  B  is  twice  the  age  of  C  ;  A  is  45  yr.  older  than 
C  :  what  is  the  age  of  each  ? 

10.  A  has  ^-  as  much  money  as  B ;  B  has  |  as  much 
as  C  ;  C  has  $15  more  than  A :  how  much  money  has  each? 

11.  A  farmer's  sheep  are  in  3  fields;  the  second  con- 
tains 4  times  as  many  as  the  first;  the  third  3  times  aa 
many  as  the  second,  and  70  more  than  both  the  first  and 
second :  how  many  sheep  are  there  in  each  field  ? 

12.  The  age  of  A  is  £  the  age  of  B ;  twice  the  age  of 
A  is  |  the  age  of  C;  C  is  20  yr.  older  than  B :  what  is 
the  age  of  each  ? 


148  KAY'S  INTELLECTUAL  ARITHMETIC. 

13.  A   father,   who    had   as   many   sons  as  daughters, 
divided   818   among   them,   giving   to  each  daughter  82{ 
and  to  each  son  §1 :  how  many  children  had  he  ? 

14.  A  man  agreed  to  pay  a  laborer  82  for  every  da. 
he  worked ;  and  the  laborer,  for  every  da.  he  was  idle, 
was  to  forfeit  81 ;  at  the  expiration  of  20  da.,  the  laborer 
received  825  :  how  many  da.  was  he  idle  ? 

ANALYSIS. — Had  he  worked  every  da.  he  would  have  received, 
at  the  expiration  of  the  20  da,,  $.40;  but  as  he  received  only 
$25,  he  lost  $15  by  being  idle. 

Each  da.  he  was  idle  he  received  $3  less  than  if  he  had 
worked,  $2  wages,  and  the  $1  forfeited;  hence, 

He  was  idle  as  many  da.  as  $3  are  contained  times  in  $15, 
that  is,  5  da. 

15.  James   was   hired   for   30  da. ;    for   every   da.    he 
worked,  he  was  to   receive  30  ct.,  and  for  every  day  he 
Was  idle,  he  was  to  pay  20  ct. ;  at  the  end  of  the  time, 
he  received  85  :  how  many  da.  did  he  work  ? 

16.  When  A  and  B  entered  school,  the  age  of  A  was  3 
times  that    of  B ;  but  in  5  yr.,  A's  age  was  only  twice 
B's  :  what  were  their  ages  at  first  ? 

ANALYSIS. — Since  the  age  of  A  is  three  times  that  of  B,  and 
when  5  yr.  are  added  to  each,  the  age  of  A  is  only  twice  that 
of  B,  therefore, 

Three  times  the  age  of  B,  (which  is  the  age  of  A,)  in- 
creased by  5  yr.  =  twice  the  age  of  B  and  5  yr ;  that  is,  = 
twice  the  age  of  B,  and  twice  5  yr. ;  hence, 

The  age  of  B  must  be  5yr.;  and  that  of  A,  15  yr. 

17.  Four  yr.   ago,  the  age  of  A  was  3  times  the  age 
of  B ;  but  4  yr.  hence,  it  will  be  only  twice  his  age  :  what 
were  their  ages  4  yr.  ago  ? 

18.  B's  age  is  twice  A's ;  in  10  yr.,  A's  will  be  f  of 
B's  :  what  are  their  ages  ? 


REVIEW.— QUESTIONS.  149 

19.  A  person  has  2  watches,  and  a  chain  worth  $10 ; 
the  first  watch  and  chain  are  worth  half  as  much  as  the 
second;  the  chain  and  second  watch,  are  worth  3  times 
as  much  as  the  first :  what  is  the  value  of  each  ? 

ANALYSIS, — Since  the  1st  watch  and  chain  are  together 
worth  1-halfofthe  2nd,  the  2nd  must  be  worth  twice  the  1st  + 
twice  the  chain.  But, 

The  2nd  watch  and  chain  =  3  times  the  first;  or  the  2nd 
watch  =  3  times  the  1st  —  the  chain;  hence, 

Three  times  the  1st  —  the  chain  —  2  times  the  first  +  2  times 
the  chain;  or  the  1st  =  3  times  the  chain,  or  $30;  and 
the  2nd  =  $80. 

20.  A  person  has  2  horses,  and  a  saddle  worth  $12  : 
the  1st  horse  and  saddle  are  worth  ^  of  the  2nd  horse ; 
but  the  2nd  horse  and  saddle  are  worth  four  times  the 
1st  horse  :  what  is  the  value  of  each  horse  ? 

21.  If  a  herring  and  a  half  cost  2  pence  and  a  half, 
how  many  can  you  buy  for  9  pence  ? 


LESSON  XIIL 

1.  If  12  peaches  are  worth  84  apples,  and  8  apples 
24  plums,  how  many  plums  shall  I  give  for  5  peaches  ? 

2.  If  1  ox  is  worth  8  sheep,  and  3  oxen  are  worth 
2  horses,  what  is   the   value    of  1   horse,   if  a  sheep  is 
worth  $5  ? 

3.  A  walks  10  mi.  in  1-|  da.,  and  B  8  mi.  in  If  da. : 
how  far  will  B  travel  while  A  is  traveling  20  mi.  ? 

4.  A  bought  a  number  of  apples  at  2  for  3  ct.,  and 
as  many  more  at  2  for  5  ct. ;  he  sold  them  at  the  rate 
of  3  for  7  ct. :  how  much  per  dozen  did  he  gain  ? 

5.  C  bought  a  number  of  eggs  at  2  ct.  each,  and  twice 
as  many  at  3  ct.  each ;  he  sold  them  at  the  rate  of  3  for 
10  ct. :  how  much  per  dozen  did  he  gain  ? 

2d  Bk.  10 


150  AAY'S   INTELLECTUAL   ARITHMETIC. 

If  he  had  sold  them  at  the  rate  of  4  for  10  ct.,  how 
much  per  dozen  would  he  have  gained  or  lost  ? 

6.  Bought  a  number  of   pears  at   2  for  1  ct.,  and  as 
many  more  at  4  for  1  ct. ;  by  selling  5  for  3  ct.,  I  gained 
18  ct.  :  how  many  pears  did  I  buy  ? 

7.  A  poulterer  bought  a  number  of  ducks,  at  the  rate 
of  6  for  81,  and  twice  as  many  chickens,  at  the  rate  of 
8  for  81 ;  by  selling   2  chickens   and  1  duck  for  $£,  he 
gained  $2^ :  how  many  of  each  did  he  buy  ? 

8.  If  3  men   can   perform   a  piece   of  work  in  4  da., 
working  10  hr.  a  da.,  in  how  many  da.  can  8  men  perform 
the^same  job,  working  6  hr.  a  da.  ? 

9.  Divide  32  peaches  between  Mary,  James,  and  Lucy, 
giving  Mary  2,  and  Lucy  3  more  than  James. 

10.  If  10  gal.  of  water  per  hr.  run  into  a  vessel  contain- 
ing 15  gal.,  and  17  gal.  run  out  in  2  hr.,  how  long  will 
the  vessel  be  in  filling  ? 

11.  A  can  do  a  piece  of  work  in  4^  da.,  and  A  and  B 
together  in  2|  da. :  in  what  time  can  B  do  it  alone  ? 

12.  A,  B,  and  C,  together,  can  do  a  piece  of  work  in 
5  da. ;  A  and  B,  in  8  da. ;  and  B  and  C,  in  9  da. :  in  what 
time  can  each  of  them  do  it  alone  ? 

13.  If  5  men  or  7  women  can  do  a  piece  of  work  in 
35  da.,  in  what  time  can  5  men  and  7  women  do  it? 

14.  If  2  men  and  4  women  can  do  a  piece  of  work  in 
28  da.,  in  what  time  can  1  man  and  1  woman  do  it,  if  a 
woman  does  |  of  a  man's  work  ? 

15.  A  man  and  his  wife   consume  a  sack  of  meal  in 
15  da.;  after  living  together  6  da.,  the  woman  alone  conr 
sumed  the  remainder  in  30  da. :  how  long  would  a  ?ack 
last  either  of  them  alone  ? 

16.  A  man  had  80  eggs,  which  he  intended  to  sell  as 
follows:  36  at  3  for  4  ct.,  24  at  4  for  3  ct.,  and  the  rest 
at  10  for  17  ct. :   but  having  mixed  them,  how  must  he 
sell  them  per  dozen  to  get  the  intended  price  ? 


RE  VIEW.— QUESTIONS.  151 

17.  If  |  of  James's  money  be  increased  by  $6.  the  sum 
will    equal  what  Thomas  has;    both  together  have  $34: 
how  much  has  each  ? 

18.  If  $5  be  taken  from  the  |  of  A's  money,  the  re- 
mainder will  equal   B's ;  both  together  have  $51 :  how 
much  has  each  ? 

19.  The  age  of  A  is  twice  the  age  of  B ;  and  f  of  B's 
age  +  44  years,  =  2^  times  the  age  of  A :  what  is  the 
age  of  each  ? 

20.  A  has  not  $40 ;  but  if  he  had  half  as  many  more, 
and  $2^  besides,  he  would  have  $40  :  how  many  has  he  ? 

21.  Two  and  a  half  times  a  number  +  2£  =  100:  What 
is  the  number  ? 

22.  A  farmer  sold  |  of  his  sheep,  but  soon  afterward 
purchased  4  as  many  as   he   had  left,  when  he  had  65 
sheep :  how  many  sheep  had  he  at  first? 

23.  John  had  $50  in  silver  and  gold  j  |  of  the  silver, 
increased  by  $10,   is  equal   to   1|  times   the  gold :  what 
amount  has  he  of  each  ? 

24.  One-half  of  A's  money,  diminished  by  $3,  is  equal 
to  J-  of  B's,  increased  by  $5,  and  both  together  they  have 
$56 :  how  much  money  has  each  ? 

25.  A   started  from  C  the   same   time   that   B  started 
from   D ;  when   they  met,  |    of  the   distance  A  traveled 
=  |  of  the  distance  B  traveled ;  from  C  to  D  is  86  mi. : 
what  was  the  distance  each  traveled  ? 

26.  Two-thirds  of   A's  money  =  f   of  B's,  and  |  of 
their  difference  is  $15 :  how  much  money  has  each  ? 

27.  My  watch  and  chain  cost  f  as  much  as  my  watch ; 
3  times  the  price  of  my  chain  +  twice  the   price  of  my 
watch  =  $100  :  what  did  each  cost? 

28.  Three  towns,  A,  B,  and  C,  are  situated  on  the  same 
road ;  the  distance  from  A  to  B  is  24  mi. ;  and  |  of  the 
distance  from  A  to  B  =  f  of  the  distance  from  B  to  C  • 
how  far  is  it  from  A  to  C? 


152  KAY'S   INTELLECTUAL  ARITHMETIC. 

29.  A,  B,  and  C,  rent  a  pasture  for  §92  ;  A  puts  in  4 
horses  for  2  mon.,  B  9  cows  for  3  mon.,  and  C  20  sheep 
for  5  mon.  :  what  should  each  pay,  if  2  horses  eat  as  much 
as  3  cows,  and  3  cows  as  much  as  10  sheep  ? 

30.  John   bought   5    melons    for    5  ct.,    and   James    3 
melons  for  3  ct.  ;  they  then  joined  Thomas,  and  each  one 
ate  an  equal  part  of  the  melons  ;  when  Thomas  left,  he 
gave  them  8  ct.  :  how  should  this  be  divided  ? 

31.  A  person  having  3  sons,  A,  B,  and  C,  devised  f  of 
his  estate  to  A,  ^  to  B,  and  the  remainder  to  C  ;  the  dif- 
ference of  the  legacies  of  A  and  C  was  §160  :  what  amount 
did  each  receive  ? 

32.  The  age  of  A  is  f  of  the  age  of  B  ;  and  the  sum 
of  their  ages  -f-  half  the  age  of  B  =  twice  the  age  of 
A  —  2  yr.  :  what  is  the  age  of  each  ? 

33.  A  and  B  together  can  do  a  job   in  16  da.  ;    they 
work  4  da.,  when  A  leaves,  and  B  finishes   the  work   in 
36  da.  more  :  in  how  many  da.  can  each  do  it  ? 

34.  Three  persons,  A,  B,  and  C,  are  to  share  a  certain 
sum  of  money,  of  which  A's  part  is  $12,  which  is  %  of  the 
sum  of  the  shares  of  B  and  C;  and  |  of  C's  share  is  equal 
to  T30  of  the  sum  of  the  shares  of  A  and  B  :  what  are  the 
shares  of  each  ? 

35.  If  9  men  mow  a  field  in  12  da.,  how  many  men  can 
mow      of  it  in      of  the  time  ? 


36.  Two  men  formed  a  partnership  for  1  yr.  ;  the  1st 
put  in  §100,  and  the   2nd,   §200  :   how  much   must  the 
first  put  in  at  the  end  of  6  mon.,  to  entitle  him  to  1-half 
of  the  profits  ? 

37.  A  and  B  had  24  ct.  ;  A  said  to  B,  «  Give  me  2  of 
your  ct.,  and  I  shall  have  twice  as  many  as  you  ;"  B  re- 
plied, "  Give  me  two  of  yours,  and  I  shall  have  as  many 
as  you  :"  how  many  had  each? 

38.  A  gentleman  being  asked  his  age,  replied  :  "  The 
excess  of  f  of  50  above  *my  age,  is  equal  to  the  difference 
between  my  ag~  and  10  yr.  :"  what  was  his  age  ? 


PERCENTAGE.  153 

39.  If  I  sell  my  eggs  at  6  ct.  a  dozen,  I  will  lose  12  ct. ; 
but  if  I  sell  them  at  10  ct.,  I  will  gain  18  ct. :  what  did 
they  cost  per  dozen  ? 

40.  If  I  sell  my  sugar  at  a  certain  price  per  lb.,  I  will 
lose  $1,  but  if  I  increase  the  price  3  ct.  per  lb.,  I  will 
gain  50  ct. :  how  many  lb.  have  I  ? 

41.  If  the  labor  of  1  man  is  equal  to  that  of  2  women, 
and  the  labor  of  1  woman  is  equal  to  that  of  3  boys,  how 
many  men  would  it  take  to  do  in  1  da.,  what  12  boys  are 
a  wk.  in  doing  ? 

42.  If  sugar   worth   3i  ct.  a  lb.,  be   mixed   in   equal 
quantities  with  sugar  worth  6i  ct.,  what  will  ^  a  lb.  of 
the  mixture  be  worth,  and  how  many  lb.  must  be  given 
for  $1  ? 

43.  If  |  of  the  gain  =  7\  of  the  selling  price,  for  how 
much  will  3|  yd.  of  cloth  be  sold,  that  cost  $4  a  yd.? 

44.  When  sugar  is  worth  7  ct.  a  lb.,  a  package  was  sold 
for  24  ct.,  gaining  3  ct. :  for  how  much  should  a  package 
weighing  twice  as  much  be  sold,  to  gain  5  ct.,  when  sugar 
costs  8  ct.  a  lb.  ? 


SECTION  XXVI.— PERCENTAGE. 
LESSON  I.— GAIN  AND  LOSS. 

Explanation. — The  terras  Percentage  and  Per  cent,  mean  a  certain 
number  of  parts  out  of  each  hundred  parts ;  that  is,  a  certain 
number  of  hundredths  of  the  sum  considered. 

One  per  cent,  of  any  number  is  1-hundredth  of  that  number, 
that  is,  1  of  each  100  parts. 

Thus,  1  per  cent,  of  $100  is  $1 :  1  per  cent,  of  $200  i*  $2 ,  of 
$50,  50  cents:  of  $1,  1  cent. 

1.  What  is  1    per  cent,  of      $1  ?      $2  ?       $5  ? 

2.  What  is  2    per  cent,  of     $3  ?       $4  ?       $6  ? 


154 


RAY'S    INTELLECTUAL    ARITHMETIC. 


3.  What  is  3    per  cent,  of  $10  ?  $20  ?  $60  ? 

4.  What  is  4    per  cent,  of  $25  ?  $45  ?  $75  ? 

5.  What  is  5    per  cent,  of  $100  ?  $300  ?  $700  ? 

6.  What  is  6    per  cent,  of  $150  ?  $250  ?  $350  ? 

7.  What  is  2^  per  cent,  of  $100  ?  $200  ?  $500  ? 

8.  I  bought  a  piece  of  cloth  for  $15,  and  in  selling 
it  gained  5  per  cent,  of  the  cost :  what  did  I  gain  ? 

ANALYSIS. — 5  per  cent,  of  1  is  five  \-JiundredtTis;  5  per  cent, 
of  $1  is  5  ct. ;  of  $15  it  will  be  15  times  5  ctf.,  wfo'cA  are 
75  c£  Ansstfb  cZ. 

9.  A  grocer   bought  a   bl.   of  sugar  for  $10,  and  in 
selling  it  gained  10  per  cent.  :  how  much  did  he  gain  ? 

10.  A  farmer  having  a  flock  of   40  sheep,  lost  5  per 
cent,  of  them  :  how  many  had  he  left  ? 

11.  A  flock  of  50  sheep  increases  10  per  cent,  in  one 
year  :  how  many  are  then  in  the  flock  ? 

12.  A  lady  having  $20,  spent  10  per  cent,  for  muslin, 
and  10  per  cent,  of  the  remainder  for  calico :  how  much 
did  she  pay  for  both  ? 

One  per  cent,  of  anything  is  T^  part  of  it;  two  per 
cent,  is  T|^  =  J^ ;  four  per  cent.  T^  =  r,1-. 


5  per  cent.  =  ^ 

8|  per  cent.  =  j1^ 

10  per  cent.  =  -L 

12^  per  cent.  =  J 

16|  per  cent.  =   1 


20  per  cent,  = 
25  per  cent.  = 
33|  per  cent.  = 
50  per  cent.  = 
75  per  cent.  = 


13.  I  paid  30  ct.   per  yd.  for   muslin :    at  what  price 
must  I  sell  it,  to  make  10  per  cent.  ? 

ANALYSIS. —  Ten  per  cent,  being  IQ-hundredths,  or  \-tenth, 
I  must  add  to  the  first  cost  \-tenih  of  itself:  but, 


PERCENTAGE.  155 

One-tenth  of3Qct.  is  3  ct,  and  30  ct.  adaed  to  3e£.  arc 
33  ct  Ans.  33  ct.  per  yd. 

14.  To  make  12|  per  cent,  profit,  what  must  muslin  be 
sold  at  that  cost  8  ct.  per  yd.?     25  ct.  ? 

15.  To  make    8|  per  cent,  profit,  what  must  sugar  be 
sold  for  that  cost  6  ct.  per  Ib.  ?     12  ct.  ? 

16.  To  make  25  per  cent,  profit,  what  must  calico  be 
sold  for  that  cost  12  ct.  per  yd.?    16  ct?   20  ct.?    35  ct? 

LESSON   II. 

1.  A  merchant  bought  cloth  at  $5  per  yd.,  and  sold 
it  at  $7  per  yd. :  what  did  he  gain  on  a  yd.  ?     how  much 

v  per  cent.  ? 

ANALYSIS. — Since  Tie  bought  at  $5  and  sold  at  $7  per  yd.j 
he  gained  $2  on  every  $5,  that  is, 

He  gained  |  of  the  first  cost ;  ^  of  this  is  T^j,  and  §  are 
twice  -£$j  which  are  T4^,  or  40  per  cent. 

2.  James  bought  a  melon  for  4  ct.,  and  sold  it  for  5  ct. : 
what  per  cent,  did  he  gain  ? 

3.  An  orange  was  bought  for  5  ct.,  and  sold  for  4  ct. : 
what  was  the  per  cent,  of  loss  ? 

4.  Thomas  bought  a  watch  for  $4,  and  sold  it  for  $6 : 
what  per  cent,  did  he  gain  ? 

5.  Henry  bought  a  horse  for  $15,  and  sold  it  for  $24 : 
what  per  cent,  did  he  gain  ? 

6.  A  keg  of  wine  holding  5  gal.,  lost  6  qt  by  leak- 
age :  what  was  the  loss  per  cent.  ? 

7.  By  selling  citrons  at  6  ct.  each,  John  cleared  1  of 
the  first  cost :  what  per  cent,  would  he  have  cleared   by 
selling  them  at  8  ct.  each  ? 

8.  A  merchant  bought  cloth  at  the  rate  of  6  yd.  for 
83,  and  sold  it  at  the  rate  of  5  yd.  for  $4 :  what  per  cent, 
did  he  gain  ? 


156  RAY'S   INTELLECTUAL   ARITHMETIC. 

9.  Henry  sold  melons  at  8  ct.  each,  and  lost  i  of  the 
first  cost :  what  per  cent,  would  he  have  lost  by  selling 
them  at  3  for  25  ct. :  what  per  cent  would  he  have  gained 
by  selling  them  at  2  for  25  ct.  ? 

10.  James  bought  a  lot  of  lemons,  at  the  rate  of  2  for 
3  ct. ;  but  finding  them  damaged,  he  sold  them  at  the  rate 
of  3  for  2  ct. :  what  per  cent,  did  he  lose  ? 

11.  Sold  a  watch  for  $12,  and  gained  20  per  cent.:  what 
was  the  first  cost  ? 

ANALYSIS. — 20  per  cent,  is  -%§$  or  £,  hence  the  gain  was  equal 
to  \  of  the  cost;  therefore, 

The  watch:*oldfor  |  +  I  =  f  of  the  cost  •  12  then  is  f  of 
what? 

12.  I  sold  a  piece  of  cloth  for  S26,  and  gained  30  per 
cent. :  what  did  the  cloth  cost  me  ? 

-   13.  If  there  is  a  gain  of  40  per  cent,  when  muslin  is 
sold  at  14  ct.  a  yd.,  what  is  the  cost  price  ? 

14.  By  selling  a  horse  for  §45,  there  was  a  gain  of  12^ 
per  cent. :  what  did  the  horse  cost  ? 

15.  Sold  a  horse  for  145,  and  lost  10  per  cent. :  what 
was  the  cost  ? 

16.  Thomas  sold  a  watch  for  §21,  and  gained  75  per 
cent. :  what  did  he  pay  for  it  ? 

17.  James  sold  10  oranges  for  40  ct.,  and  gained  33^ 
per  cent. :  how  much  did  each  orange  cost  ? 

18.  "When  an  article  is  sold  at  |  of  its  cost,  what  is  the 
gain  per  cent.  ? 

19.  When  an  article  is  sold  at  |  of  its  cost,  what  is  the 
loss  per  cent.?     at  f  ?     at  T%  ?     at  $%  ? 

LESSON   III. 

1.  When  the  gain  is  20  per  cent.,  what  part  of  the 
cost  is  equal  to  the  gain  ?  when  it  is  75  per  cent.  ? 


PERCENTAGE.  157 

2.  When  the  gain  is  100  per  cent.,  what  part  of  the 
cost  is  equal  to  the  gain  ?     when  it  is  150  per  cent.  ? 

3.  When   the  loss   is  25  per  cent.,  what  part  of  the 
cost  is  equal  to  the  loss  ?     when  it  is  35  per  cent.  ? 

4.  What  is  the  loss  per  cent,  when  the  whole  is  lost? 
What  is  the  gain  per  cent,  when  the  gain  is  three  times 
the  cost? 

5.  When  -i  of  the  gain  is  equal  to  \  of  the  cost,  what 
is  the  gain  per  cent.  ? 

Explanation. — If  1-third  of  the  gain  is  equal  to  1-fifth  of  the 
cost,  the  whole  gain  is  equal  to  3-fifths  of  the  cost,  and  3-fifths 
are  60-hundredths  or  60  per  cent. 

6.  A  sold  a  watch,   so  that  §  of  the   gain  was  equal 
to  26-  of  the  cost :  what  did  he  gain  per  cent.  ? 

7.  When  |  of  the  gain  is  equal  to  i|  of  the  cost,  what 
is  the  gain  per  cent.  ? 

8.  Sold  a  watch  for  $10,  by  which  I  gained  25  per 
cent. :  what  per  cent,  would  I  have  gained  by  selling  it 
for  $12? 

9.  By  selling  muslin  at  7  ct.  per  yd.,  there  is  a  loss 
of  124  per  cent. :  what  will  be  the  loss  per  cent,  by  sell- 
ing it  at  6  ct.  per  yd.  ? 

10.  By  selling  my  horse   for   $35,    there  was  a   loss 
of  16|  per  cent. :  what  would  have  been  the  gain  per  cent, 
by    selling  him  for  $63? 

11.  I  bought  a  watch  for  $18,  which  was  20  per  cent, 
more  than  its  value  :  I  sold  it  at  10  per  cent,  less  than  its 
value  :  what  sum  did  I  lose  ? 

12.  A  sold  B  a  watch  for  $60,  and  gained  20  per  cent. : 
afterward  B  sold  it  and  lost  20  per  cent,  on  what  it  cost 
him  :  how  much  did  B  lose  more  than  A  gained  ? 

13.  A   watchmaker  sold   2  watches  for  $30  each :   on 
one  he  gained  25  per  cent.,  and  on  the  other  he  lost  25 
per  cent. :  how  much  did  he  gain  or  lose  by  the  sale  ? 


158  RAY'S   INTELLECTUAL   ARITHMETIC. 

14.  By  selling  4  apples  for  3  ct.,  a  dealer  gains  50  pei 
cent. :  what  per  cent,  will  he  gain  by  selling  them  at  the 
rate  of  5  for  4  ct.  ? 

15.  Sold  5  lemons  for  4  ct.,  and  lost  20  per  cent. :  what 
per  cent,  will  I  lose  by  selling  6  for  5  ct.  ? 

16.  Two-thirds  of  10  per  cent,  of  60,  is  i  of  what  per 
cent,  of  40  ? 

17.  One-half  of  f  of  50  per  cent,  of  120,  is  10  less 
than  20  per  cent,  of  what  ? 


iv.— INTEREST. 

EXPLANATION. — Interest  is  money  paid  for  the  use  of  money. 
The  Principal  is  the  sum  of  money  which  is  loaned. 
The  Amount  is  the  principal  and  interest  added  together. 
The  Rate  Per  Cent,  is  so  many  cents  paid  on  each  dollar. 

1.  If  the  interest  of  $1  at  6  per  cent,  for  1  yr.,  is  6  ct., 
what  will  be  the  interest  of  $10?     of  $12?     of  $15? 
of  $20  ? 

2.  What  is  the  interest  of  $2  for  3  yr.,  at  5  per  cent.? 

ANALYSIS. — The  interest  of  $1  for  1  yr.  at  5  per  cent.,  is  5 
ct. ;  and  for  $2  the  interest  is  twice  as  much  as  for  $1,  which 
is  2  times  5  ct.,  equal  1 0  ct. ;  and, 

For  3  yr.  the  interest  is  3  times  as  much  as  for  1  yr.,  which 
is  3  times  10  ct.,  equal  30  ct.  Ans.  30  ct. 

3.  Find  the  interest  of    $5  for  2  yr.,  at  6  per  cent. 

4.  Find  the  interest  of    $8  for  5  yr.,  at  5  per  cent. 

5.  Find  the  interest  of  $20  for  3  yr.,  at  8  per  cent. 

6.  Find  the  interest  of  $25  for  6  yr.,  at  4  per  cent. 

7.  Find  the  interest  of  $40  for  4  yr.,  at  5  per  cent. 

8.  Find  the  interest  of  $50  for  3  yr.,  at  6  per  cent. 


PERCENTAGE.  159 

9.   Find  the  interest  of  $60  for  2  yr.,  at  7  per  cent. 
10.   Find  the  interest  of  $75  for  3  jr.,  at  4  per  cent. 

LESSON  Y. 

1.  What  the  interest  of  $50  for  5  mon.,  at  6  per  cent.? 

ANALYSIS. — For  5  mon.  the  interest  will  be  5  times  as  much 
a#for\  mon.;  and  for  1  mon.,  \-twelfth  as  much  as  for  a  yr. 

2.  Find  the  interest  of  $60  for  4  mon.,  at  5  per  cent. 

3.  Find  the  interest  of  $80  for  7  mon.,  at.6;per  cent. 

4.  Find  the  interest  of  $40  for  9  mon.,  at  8  per  cent. 

5.  Find  the  interest  of  $75  for  8  mon.,  at  9  per  cent. 

What  is  the  interest 

6.  Of  $120  for    6  mon.  15  da.,  at  5  per  cent.  ? 

7.  Of  $150  for  10  mon.  10  da.,  at  4  per  cent.  ? 

8.  Of    $45  for  11  mon.  23  da.,  at  8  per  cent.  ? 

9.  Of  $200  for    4  mon.  24  da.,  at  6  per  cent.  ? 

10.  Of  $480  for  9  mon.  18  da.,  at  5  per  cent.? 

11.  Of  $360  for  5  mon.  19  da.,  at  5  per  cent.? 

12.  Of  $144  for  8  mon.  25  da.,  at  4  per  cent.? 

13.  Of    $40  for  1  yr.  4  mon.,    at  6  per  cent.  ? 

14.  Of    $60  for  2  yr.  3  mon.,     at  5  per  cent.  ? 

15.  Of    $75  for  1  yr.  3  mon.  6  da.,  at  4  per  cent.  ? 

What  is  the  amount 

16.  Of  $25  for  3  yr.,  at  4  per  cent.  ? 

17.  Of  $40  for  2  yr.,  at  5  per  cent.  ? 

18.  Of  $55  for  3  yr.,  at  8  per  cent.  ? 

19.  Of  $30  for  1  yr.  4  mon.,  at  7  per  cent.  ? 


160  RAY'S   INTELLECTUAL   ARITHMETIC. 

20.  Of  $50  for  2  yr.  3  mon.  6  da.,  at  6  per  cent.  ? 

21.  Of  §90  for  1  yr.  3  inon.  6  da.,  at  8  per  cent.  ? 

LESSON  VI. 

1.  The  interest  of  a  certain  principal  for  2  yr.,  at  6 
per  cent.,  is  $3 :  what  is  the  principal  ? 

ANALYSIS. — The  interest  of  $1  for  2  yr.,  at  6  per  cent.,  is 
12  ct. ;  and  the  principal  must  be  as  many  times  $1  as  12  ct. 
are  contained  times  in  $3.  Ans.  $25. 

2.  The  interest  of  a  certain  principal  for  3  yr.,  at  4 
per  cent.,  is  $6  :  what  is  the  principal  ? 

3.  What  principal  at  interest  for  4  yr.,  at  5  per  cefct., 
will  produce  $12  interest  ? 

4.  What  principal  at  interest  for  5  yr.,  at  8  per  cent., 
will  produce  $30  interest? 

5.  What  principal  at  interest  for  4  yr.,  at  7J  per  cent., 
will  produce  $42  interest  ? 

6.  What  principal   at  interest  for  2  yr.  6  mon.,  at  6 
per  cent.,  will  produce  $36  interest? 

7.  What  principal  at  interest  for  3  yr.  4  mon.,  at  6 
per  cent.,  will  produce  $70  interest  ? 

8.  A   father  wishes   to   place  such  a  sum  at  interest 
at  5  per  cent.,  as  will  produce  for  his  son  an  annual  in- 
come of  $200  :  what  sum  must  he  invest  ? 


LESSON   VII. 

1.  What  principal  on  interest  for  2  yr.,  at  5  per  cent., 
will  amount  to  $55  ? 

ANALYSIS. —  The  amount  of  $1  for  2  yr.  at  5  per  cent.,  is 
$1.10,  and  it  will  require  as  many  times  $1  to  amount  to  $55 
as  $1.10  is  contained  times  in  $55. 


PERCENTAGE.  161 

What  principal  on  interest, 

2.  At    6  per  cent.,  for  3  yr.,  will  amount  to  $236  ? 

3.  At    5  per  cent.,  for  4  yr.,  will  amount  to  $600  ? 

4.  At  10  per  cent.,  for  5  yr.,  will  amount  to  $375  ? 

5.  At    6  per  cent.,  for  5  yr.,  will  amount  to  $390  ? 

6.  The  amount  due  on  a  note  which  has  been  on  in- 
terest 3  yr.  4  mon.,  at  6  per  cent.,  is  $30 :  what  is  the 
face  of  the  note  ? 

7.  Two-fifths   of  A's  money  on  interest  for  2  yr.  6 
mon.,  at  8  per  cent.,  is  $60:  what  is  his  whole  money  ? 


LESSON   VIII. 

1.  In  what  time,  at  6  per  cent.,  will  $50  give  $10 
interest  ? 

ANALYSIS. — The  interest  of  $50  for  1  yr.,  at  6  per  cent,  is 
$3 ;  and  it  will  require  $50  as  many  yr.,  to  give  $10  interest, 
as  $3  is  contained  times  in  $10,  which  is  3J. 

In  what  time, 

2.  At    5  per  cent.,  will    $40  give    $8  interest  ? 

3.  At    8  per  cent.,  will    $75  give  $15  interest? 

4.  At  10  per  cent.,  will    $60  give  $16  interest? 

5.  At    5  per  cent.,  will  $14$  give  $24  interest  ? 

6.  At    6  per  cent,  will    $25  give  $10  interest? 

7.  In  what  time,  at  4  f>er  cent.,  will  any  given  prin- 
cipal double  itself? 

ANALYSIS. — At  1  per  cent,  any  given  sum,  as  $100,  will 
double  itself  in  100  yr. ;  and, 

At  4  per  cent,  it  will  double  itself  in  J  of  the  time  that  it 
will  at  1  per  cent :  \  of  lOOyr.  is  25  yr.  Ans.  25  yr. 


162  KAY'S  INTELLECTUAL  ARITHMETIC. 

8.  In  what  time  will  any  given  principal  double  itself, 
at  2  per  cent.  ?     at  3  per  cent.  ?     at  5  per  cent.  ?     at  6  per 
cent.  ?     at  7  per  cent.  ?     at  8  ?     at  10  ?     at  12  ? 

9.  In  what  time  will  any  given  principal  treble  itself, 
at  5  per  cent.  ? 

10.  In  what  time  will  any  given  principal  treble  itselfj 
at  8  per  cent.  ?     at  10  per  cent.  ? 


LESSON  IX. 

1.  At  what  per  cent,  will  §200,  in  2  yr.,  give  $24  in. 
terest  ? 

ANALYSIS. — At  I  per  cent,  for  2  yr.,  $200  will  give  $4  in* 
terest;  and, 

It  will  take  as  many  times  1  per  cent,  for  $200  to  give  $24 
interest,  as  $4,  the  interest  of  $200  at  1  per  cent.,  is  contained 
times  in  $24. 

At  what  per  cent., 

2.  Will    $50  in  5  yr.,  give  $20    interest  ? 

3.  Will    $75  in  3  yr.,  give  $11J  interest  ? 

4.  Will  $300  in  3  yr.,  give  $63    interest? 

5.  Will  $300  in  2  yr.  3  mon.,  give  $54  interest  ? 

6.  Will  $240  in  3  yr.  4  mon.,  give  $56  interest? 

7.  Will  $200  in  4  yr.,  amount  to  $240  ? 

8.  Will  $150  in  3  yr.  8  mon.,  amount  to  $183  ? 

j,        9.   Will  any  given  principal  double  itself  in  20  yr.  ? 

ANALYSIS. — Any  principal,  as  $100,  will  double  itself 
in  1  yr.,  at  100  per  cent.,  and  in  20  yr.,  at  ^  of  100  per 
cent.,  or  5  per  cent. 

At  what  per  cent., 

10.    Will  any  given  principal  double  itself  in  12  yr.  ? 


PERCENTAGE.  163 

11.  Will  any  given  principal  double  itself  in  10  yr.  ? 

12.  Will  any  given  principal  double  itself  in     8  yr.  ? 
in  5  yr.  ?     in  4  yr.  ?     in  2  yr.  ? 


LESSON  X. 

1.  What  principal,  at  5  per  cent,  for  4  yr.,  will  amount 

to  $72? 

ANALYSIS. — $1,  at  5  per  cent.,  for  4  yr.,  will  amount  to 
$1.20;  and, 

The  required  principal  will  be  as  many  times  $1,  as  the 
amount  of  $1  at  the  given  rate  for  the  given  time,  which  is 
$1.20,.&  contained  times  in  $72. 

2.  What  principal,   at   6   per   cent.,   for    5  yr.,    will 
amount  to  $520  ? 

3.  What  principal,   at  4   per    cent.,    for   5  yr.,   will 
amount  to  $30  ? 

4.  What  principal,   at   10  per   cent.,  for  5  yr.,  will 
amount  to  $750  ? 

5.  What   principal,    at   5    per    cent.,   for   3  yr.,   will 
amount  to  $345  ? 

6.  What   principal,    at   6    per    cent.,   for   4  yr.,   will 
amount  to  $496  ? 

7.  What  is  the  present  worth  of  $24,  due  4  yr.  hence> 
reckoning  interest  at  5  per  cent.  ? 

Explanation. — The  present  worth  is  that  principal  of  which  $24 
is  the  amount.  The  discount  is  the  interest  on  the  present  worth. 

8.  What  is  the  present  worth  of  $65,  due  5  yr.  hence, 
interest  at  6  per  cent.  ?     what  the  discount  ? 

9.  What  is  the  present  worth  of  $55,  due  5  yr.  hence, 
interest  at  5  per  cent.  ?     what  the  discount  ? 

10.  A  owes  $77,  payable  6  yr.  8  mon.  hence :  what  will 
he  gain  by  paying  it  now,  money  worth  6  per  cent.  ? 


164  RAY'S  INTELLECTUAL  ARITHMETIC, 


LESSON  XI. 

1.  At  6  per  cent.,  for  4  yr.  2  mon.,  what  part  of  the 
principal  is  equal  to  the  interest  ? 

2.  At  5  per  cent.,  for  5  yr.,  what  part  of  the  amount 
is  equal  to  the  interest  ? 

3.  When  the  interest  for  2  yr.  =  1  of  the  principal, 
what  is  the  rate  per  cent.  ? 

4.  When   the   interest  for   2  yr.    6  mon.  ==  \    of   the 
principal,  what  is  the  rate  per  cent.  ? 

5.  When  the  interest,  at  10  per  cent.  =  f  of  the  prin- 
cipal, what  is  the  time  ? 

6.  When  3  times  the  yearly  interest  =  ^  of  the  prin- 
cipal, what  is  the  rate  per  cent.  ? 

7.  When  I  of  the  interest  for  2  yr.  =  ^  of  the  prin- 
cipal, what  is  the  rate  per  cent,  ? 

8.  When  f  of  the  interest  for  3  yr.  =  g%  of  the  prin- 
cipal,  what  is  the  rate  per  cent.  ? 

9.  The   interest   for  8  mon.   is    ^   of   the  principal : 
what  is  the  interest  of  §200  for  1  yr.  4  m$n.  ? 

10.  If  the  interest  for  1  yr.  4  mon.,  is  ^  of  the  prin- 
cipal, what  the  interest  of  $100  for  1  yr.  8  mon.  18  da.  ? 

11.  In  what  time  will  any  principal  at  5  per  cent.,  give 
the  same  interest  as  in  4  yr.,  at  10  per  cent.  ? 

12.  The  interest  of  A's  and  B's  money  for  3|  yr.,  at  5 
per  cent.,  is  $40,  and   A's  money   is  twice  that  of  B's : 
what  sum  has  each  ? 

13.  Twice  A's  money  =  3  times  B's ;  and  the  interest 
at  7  per  cent,  for  1|  yr.,  of  what  they  both  have,  is  $49 : 
how  much  money  has  each  ? 

14.  One-half  of  A's  money  =  f  of  B's ;  and  the  inter- 
est of  |  of  A's  and  ^  of  B's  money,  at  4  per  cent,  for 
2  yr.  3  mon.,  is  $18 :    how  much  has  each  ? 

THE    END, 


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